Sequences and Limits of Sequences
In a video game, Malik's character stands 30 feet from a door. Each turn he moves half of the remaining distance. Will he ever reach the door? How close can he get?
That question is the heart of calculus: what happens to a list of numbers as you go farther and farther along it? On this page you will learn to find these limits quickly, to prove them, and to handle sequences defined in terms of themselves. Limits of functions, derivatives and integrals are all built on this idea.
What does the limit of a sequence mean?
A sequence is a list of numbers $a_1, a_2, a_3, \dots$ that is, a function whose inputs are the positive integers $n = 1, 2, 3, \dots$. The number $a_n$ is the $n$th term.
Here are the first eight terms of $a_n = \dfrac{1}{n^2}$. They fall toward the horizontal axis, but never touch it.
Two families of sequences come up constantly, so memorize them.
The widget above is Malik's door problem: the distance left before turn $n$ is $30\left(\tfrac12\right)^{n-1}$ feet. Since $r=\tfrac12$ has $|r|<1$, the distance tends to $0$. He never reaches the door, but he gets as close as you like. That is exactly what $\lim = 0$ means.
Given the sequence whose general term is $a_n = \frac{1}{n^2}$, write the first four terms and reason, intuitively, what the value of its limit is as $n$ tends to…
Given the sequence whose general term is $a_n = \frac{1}{n^2}$, write the first four terms and reason, intuitively, what the value of its limit is as $n$ tends to infinity:
$$\lim_{n \to \infty} a_n$$It is estimated that the value of a laptop after $n$ years is given by the sequence $v_n = 1200 \cdot (0,8)^n$. Calculate the limit of this sequence and interpret what…
It is estimated that the value of a laptop after $n$ years is given by the sequence $v_n = 1200 \cdot (0,8)^n$. Calculate the limit of this sequence and interpret what the result means in the context of the depreciation of the equipment.
$$\lim_{n \to \infty} 1200 \cdot (0,8)^n$$Rational sequences: compare the degrees
Most sequences you will meet are quotients of polynomials in $n$, like $\dfrac{5n^2-3n+1}{2n^2+7}$. Both top and bottom go to $\infty$, which looks like the indeterminate form $\dfrac{\infty}{\infty}$. The trick is to see which power of $n$ is the boss.
Determine the value of the limit of the sequence $b_n$ as $n$ grows indefinitely: $$b_n = \frac{100n + 50}{n^2 - 1}$$
Determine the value of the limit of the sequence $b_n$ as $n$ grows indefinitely:
$$b_n = \frac{100n + 50}{n^2 - 1}$$Calculate the limit of the following rational sequence by comparing the degrees of the numerator and the denominator: $$a_n = \frac{5n^2 - 3n + 1}{2n^2 + 7}$$
Calculate the limit of the following rational sequence by comparing the degrees of the numerator and the denominator:
$$a_n = \frac{5n^2 - 3n + 1}{2n^2 + 7}$$On a social network, the number of "likes" of a post in the first few hours follows the progression $c_n = \frac{4n^2}{n+2}$, where $n$ is the number of hours since…
On a social network, the number of "likes" of a post in the first few hours follows the progression $c_n = \frac{4n^2}{n+2}$, where $n$ is the number of hours since publication. What happens to the amount of "likes" in the very long term? Calculate:
$$\lim_{n \to \infty} \frac{4n^2}{n+2}$$Polynomials in disguise
Sometimes the polynomials are not written in expanded form, or the sequence is a difference of two quotients (the form $\infty-\infty$). You still use the same idea.
Factored forms. Don't expand. Only the leading term of each factor matters. For $\dfrac{(2n+1)^2(n-3)}{(n+4)^3+7n}$, the numerator behaves like $(2n)^2\cdot n=4n^3$ and the denominator like $n^3$. To be rigorous, divide everything by $n^3$: $$\frac{\left(2+\frac1n\right)^2\left(1-\frac3n\right)}{\left(1+\frac4n\right)^3+\frac{7}{n^2}}\ \longrightarrow\ \frac{4\cdot1}{1}=4.$$
A difference of fractions. $\infty-\infty$ is indeterminate, so first combine into one fraction.
Finding parameters. If a limit must equal $0$ and the expression has the form $\frac{\text{polynomial}}{\text{polynomial}}-an-b$, use polynomial division to split off the "linear part".
Calculate the limit of the following sequence by analysing the degrees of the polynomials and justifying the result by comparing the terms of highest degree:…
Calculate the limit of the following sequence by analysing the degrees of the polynomials and justifying the result by comparing the terms of highest degree:
$a_n = \frac{(3n - 1)^2 \cdot (n + 2)}{(2n + 3)^3 - 5n}$
Calculate the limit of the following sequence, which presents a subtraction of fractions with different denominators:…
Calculate the limit of the following sequence, which presents a subtraction of fractions with different denominators:
$a_n = \frac{2n^2 + 1}{n - 1} - \frac{2n^3 + 5}{n^2 + 2}$
Indicate whether the sequence is divergent or convergent after solving the indeterminate form $\infty - \infty$.
Find the values of the real parameters $a$ and $b$ so that the following equality between the limit of the sequence and the indicated value is satisfied: $$\lim_{n \to…
Find the values of the real parameters $a$ and $b$ so that the following equality between the limit of the sequence and the indicated value is satisfied:
$$\lim_{n \to \infty} \left( \frac{n^2 + 5n + 1}{n - 2} - an - b \right) = 0$$Square roots and the conjugate
A root is a power with a fractional exponent, so the "degree" idea still works. For $n>0$, $\sqrt{n^2}=n$, so $\sqrt{9n^2+n}$ behaves like $3n$. In general, a root with index $m$ of a polynomial of degree $k$ acts like a term of degree $\frac km$. Then compare degrees as in the previous sections.
When you have a difference of roots, $\infty-\infty$ again, multiply and divide by the conjugate: $(A-B)(A+B)=A^2-B^2$ removes the root.
Roots with different indices. There is no single conjugate for $\sqrt[3]{\ }-\sqrt{\ }$. Add and subtract the "middle term" $n$ to split the difference into two pieces, each with a conjugate of its own type.
Calculate the limit of the following sequence involving a square root, by analysing the term of highest degree: $$d_n = \frac{\sqrt{9n^2 + n}}{2n - 3}$$
Calculate the limit of the following sequence involving a square root, by analysing the term of highest degree:
$$d_n = \frac{\sqrt{9n^2 + n}}{2n - 3}$$Solve the limit of the sequence $a_n = \sqrt{4n^2 + 3n - 1} - 2n$. To visualize the behavior of the sequence, observe the following diagram where the difference between…
Solve the limit of the sequence $a_n = \sqrt{4n^2 + 3n - 1} - 2n$. To visualize the behavior of the sequence, observe the following diagram where the difference between the curve of the root and its linear asymptote is represented:
Determine the limit of the following sequence, which involves a difference of roots of different indices (cube and square). To solve it, you must force the appearance of…
Determine the limit of the following sequence, which involves a difference of roots of different indices (cube and square). To solve it, you must force the appearance of a common factor or use the technique of adding and subtracting an intermediate term that allows the expression to be separated into two manageable limits:
$$\lim_{n \to \infty} \left( \sqrt[3]{n^3 + 3n^2} - \sqrt{n^2 - 4n} \right)$$Growth rates: who wins at infinity?
The sequences $n^2$, $2^n$ and $n!$ all go to $\infty$, but not at the same speed. Compare them:
| $n$ | $\ln n$ | $n^2$ | $2^n$ | $n!$ | $n^n$ |
|---|---|---|---|---|---|
| 10 | 2.30 | 100 | 1,024 | $3.6\times10^{6}$ | $10^{10}$ |
| 20 | 3.00 | 400 | $1.05\times10^{6}$ | $2.4\times10^{18}$ | $1.05\times10^{26}$ |
| 30 | 3.40 | 900 | $1.07\times10^{9}$ | $2.7\times10^{32}$ | $2.1\times10^{44}$ |
The widget shows $\dfrac{n^3}{2^n}$. It rises at first (a cubic starts out ahead), but the exponential eventually takes over and the ratio falls to $0$.
To use the hierarchy, factor out the dominant term in the top and the bottom.
Using the comparison of orders of infinity (exponential versus polynomial), find the limit of the sequence: $$e_n = \frac{3^n}{n^{10} + 500}$$
Using the comparison of orders of infinity (exponential versus polynomial), find the limit of the sequence:
$$e_n = \frac{3^n}{n^{10} + 500}$$Given the following sequences, use the hierarchy of infinities to determine, without performing extensive algebraic operations, the value of their limit as…
Given the following sequences, use the hierarchy of infinities to determine, without performing extensive algebraic operations, the value of their limit as $n \to \infty$:
- a) $b_n = \frac{n^5 + 10^6}{\ln(n^2 + 1) + 0.5^n}$
- b) $c_n = \frac{\sqrt{n^3 + 4n}}{2n + \sqrt[3]{n^5}}$
- c) $d_n = \frac{1.02^n}{n^{1000} + 5n}$
In the study of computing algorithms, it is essential to know the speed at which certain functions grow. Calculate the limit of the following sequence by comparing the…
In the study of computing algorithms, it is essential to know the speed at which certain functions grow. Calculate the limit of the following sequence by comparing the orders of infinity of the different terms (factorials, exponentials and powers):
$$\lim_{n \to \infty} \frac{n! + 10^n}{n^n - 5^n}$$Explain with reasoning which term dominates in the numerator and which in the denominator to justify your result.
The Squeeze Theorem
Sometimes a sequence is messy, but it is trapped between two simple sequences that head to the same place. Then it has no choice.
Squeezing a sum. For a sum with $n$ terms, bound every term by the smallest and the largest one.
In a biotechnology laboratory, the number of bacteria of a strain (in millions) after $n$ hours is modelled by the sequence:…
In a biotechnology laboratory, the number of bacteria of a strain (in millions) after $n$ hours is modelled by the sequence:
$P_n = \frac{12n^2 + \text{sen}(n)}{3n^2 + 500}$
Determine the bacteria population as time tends to infinity. Reason what happens to the term $\text{sen}(n)$ in the calculation of the limit.
Use the Squeeze Theorem (or Sandwich Theorem) to find the value of the limit of the sequence $a_n$ defined by the following sum of fractions: $$an = \frac{n}{n^2+1} +…
Use the Squeeze Theorem (or Sandwich Theorem) to find the value of the limit of the sequence $a_n$ defined by the following sum of fractions:
$$a_n = \frac{n}{n^2+1} + \frac{n}{n^2+2} + \frac{n}{n^2+3} + \dots + \frac{n}{n^2+n}$$The number e
Some limits settle on a famous irrational number. Try $\left(1+\frac1n\right)^n$: the base gets closer to $1$, but the exponent gets bigger.
| $n$ | 1 | 10 | 100 | 1,000 | 1,000,000 |
|---|---|---|---|---|---|
| $\left(1+\frac1n\right)^n$ | 2 | 2.5937 | 2.7048 | 2.7169 | 2.71828 |
Why the second form? $\left(1+\frac1n\right)^{kn}=\left[\left(1+\frac1n\right)^n\right]^k\to e^k$, because the limit of a power is the power of the limit.
Identify which of the following expressions corresponds to the definition of the number $e$ and calculate its limit: $$x_n = \left(1 + \frac{1}{n}\right)^{2n}$$
Identify which of the following expressions corresponds to the definition of the number $e$ and calculate its limit:
$$x_n = \left(1 + \frac{1}{n}\right)^{2n}$$The indeterminate form $1^\infty$
When a base tends to $1$ and the exponent tends to $\infty$, the limit is not simply "$1^\infty=1$". The size of the exponent matters, so there is a recipe that comes straight from the definition of $e$.
An investor analyses the growth of a cryptocurrency whose daily evolution follows the law of an exponential sequence. The relative value after $n$ days is given by:…
An investor analyses the growth of a cryptocurrency whose daily evolution follows the law of an exponential sequence. The relative value after $n$ days is given by:
$V_n = \left( \frac{5n^2 + 3n}{5n^2 - 2} \right)^{n+1}$
Calculate the value to which the currency tends in the long term using the relationship with the number $e$.
Find the value of the real parameters $k$ and $t$ so that the following equality is verified: $\lim_{n \to \infty} \left( \frac{kn + 5}{3n - 1} \right)^n = e^t$ Knowing…
Find the value of the real parameters $k$ and $t$ so that the following equality is verified:
$\lim_{n \to \infty} \left( \frac{kn + 5}{3n - 1} \right)^n = e^t$
Knowing that the limit of the base is $1$ and that the final result is $e^2$.
Calculate the value of the following limit of a sequence that presents an indeterminate form of the type $1^\infty$, paying special attention to the algebraic…
Calculate the value of the following limit of a sequence that presents an indeterminate form of the type $1^\infty$, paying special attention to the algebraic manipulation of the base and the exponent:
$$\lim_{n \to \infty} \left( \frac{2n^2 + 5n - 1}{2n^2 - 3n + 2} \right)^{\frac{n^2 + 1}{2n - 3}}$$Recursive sequences
Some sequences are defined by a rule that builds each term from the previous one: $a_{n+1}=f(a_n)$. If you know the sequence converges, you can find its limit without computing a single extra term.
The same shift trick lets you find the explicit term and the partial sums of a recursive sequence. This is what a "total distance" problem needs.
Study the behavior of the recursive sequence defined by: $a_1 = 2$ $a_{n+1} = \frac{1}{2} \left( a_n + \frac{4}{a_n} \right)$ If we assume that the sequence is…
Study the behavior of the recursive sequence defined by:
$a_1 = 2$
$a_{n+1} = \frac{1}{2} \left( a_n + \frac{4}{a_n} \right)$
If we assume that the sequence is convergent and that its limit is $L$ ($L > 0$), calculate this value by solving the equation $L = \frac{1}{2} \left( L + \frac{4}{L} \right)$. Which famous irrational constant does this result correspond to?
A sequence is defined recursively by the formula: $$a_1 = \sqrt{6}$$ $$a_{n+1} = \sqrt{6 + a_n}$$ Assuming that the sequence is convergent (you can visualize its…
A sequence is defined recursively by the formula:
$$a_1 = \sqrt{6}$$ $$a_{n+1} = \sqrt{6 + a_n}$$Assuming that the sequence is convergent (you can visualize its behavior in the diagram below), calculate the exact value of its limit $L = \lim_{n \to \infty} a_n$.
A surveillance drone is programmed to make a series of jumps in a straight line. In the first second it travels $d$ feet. In each subsequent second, it travels a…
A surveillance drone is programmed to make a series of jumps in a straight line. In the first second it travels $d$ feet. In each subsequent second, it travels a distance that is half the distance traveled in the previous second plus an extra foot to compensate for the wind. If $x_n$ is the total accumulated distance after $n$ seconds:
- a) Find the expression for the general term of the distance traveled in each second $a_n$.
- b) Determine the distance the drone will travel in a given second when $n$ tends to infinity (that is, $\lim_{n \to \infty} a_n$).
Proving a limit: the $\varepsilon$–$N$ definition
So far you found limits. Now let's prove one rigorously. The definition says that the terms stay inside any band around $L$, however narrow, from some point on.
The strategy is always the same. Start from $|an-L|<\varepsilon$, simplify the left side, and solve for $n$. The result tells you how large $N$ must be, in terms of $\varepsilon$.
Given the sequence $a_n = \frac{3n + 2}{n + 1}$, prove using the formal definition of a limit ($\varepsilon$ and $N$) that its limit is $L = 3$. To do this, for a value…
Given the sequence $a_n = \frac{3n + 2}{n + 1}$, prove using the formal definition of a limit ($\varepsilon$ and $N$) that its limit is $L = 3$.
To do this, for a value of $\varepsilon = 0.001$, determine from which term $N$ of the sequence it is satisfied that $|a_n - L| < \varepsilon$.