Convergence of a Series: Definition, Formula, Examples

For a sequence {a_n}, if we add all the members of that sequence, then we get an infinite series. Here, a₁+a₂+a₃+… is called infinite series (simply, a series) generated by the sequence {a_n}. In this article, we will study the convergence of a series along with its definition, formula, examples.

The above series is denoted by $\displaystyle \sum_{n=1}^\infty a_n$ or simply ∑a_n. That is,

$\displaystyle \sum_{n=1}^\infty a_n$ = a₁+a₂+a₃+…

Convergent Series Definition

Let us talk about the convergence of the series $\displaystyle \sum_{n=1}^\infty a_n$. Note that this series is generated by the sequence {a_n}.

Now, we consider the sequence {s_n} where

s₁ = a₁

s₂ = a₁+a₂

s₃ = a₁+a₂+a₃

$\vdots$

s_n = a₁+a₂+a₃+…+a_n.

The sequence {s_n} is called the sequence of partial sums of the series ∑a_n. The series ∑a_n is said to be convergent (or divergent) if the sequence {s_n} is convergent (or divergent).

Formula of a Series:

• For a convergent series ∑a_n, the limit $\lim\limits_{n \to \infty}$ s_n exists. If $\lim\limits_{n \to \infty}$ s_n = S, then S is the value of the series ∑a_n. In other words, S = ∑a_n = = a₁+a₂+a₃+…
• For a divergent series ∑a_n, the limit $\lim\limits_{n \to \infty}$ s_n does not exist.

Example of a Series

Discuss the convergence of the series $\dfrac{1}{1 \cdot 2}+\dfrac{1}{2 \cdot 3}+\dfrac{1}{3 \cdot 4}+\cdots$. That is, find the value of the series, if possible.

Ans: Note that the given series can be written as $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n(n+1)}$.

So $a_n=\dfrac{1}{n(n+1)}$.

Therefore,

s_n = the sum of first n-terms

= $\dfrac{1}{1 \cdot 2}+\dfrac{1}{2 \cdot 3}+\dfrac{1}{3 \cdot 4}+$ $\cdots \dfrac{1}{n(n+1)}$

= $\dfrac{2-1}{1 \cdot 2}+\dfrac{3-2}{2 \cdot 3}+\dfrac{4-3}{3 \cdot 4}+$ $\cdots \dfrac{(n+1)-n}{n(n+1)}$

= $\left( 1-\dfrac{1}{2} \right)+ \left(\dfrac{1}{2} -\dfrac{1}{3}\right)+$ $\left(\dfrac{1}{3}- \dfrac{1}{4} \right)+$ $\cdots \left(\dfrac{1}{n}-\dfrac{1}{n+1} \right)$

= $1-\dfrac{1}{n+1}$

Now, lim_n→∞ s_n = lim_n→∞ $\left( 1-\dfrac{1}{n+1} \right)$ = 1-0 =1.

So the above series converges to 1. That is, the sum of the given series is equal to 1.

Homework:

Find the sum of the following series, if possible:

1. $\dfrac{1}{2 \cdot 3}+\dfrac{1}{3 \cdot 4}+\dfrac{1}{4 \cdot 5}+\cdots$.
2. $\dfrac{1}{1 \cdot 2 \cdot 3}+\dfrac{1}{2 \cdot 3 \cdot 4}+\dfrac{1}{3 \cdot 4 \cdot 5}+\cdots$.
3. $1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\cdots$.
4. 1+2+3+4+….

Geometric Series

The series a+ar+ar²+ar³+… = $\displaystyle \sum_{n=0}^\infty ar^{n-1}$ is called the geometric series with first term = a and common ratio = r.

Common ratio = $\dfrac{\text{second term}}{\text{first term}}$ = $\dfrac{\text{third term}}{\text{second term}}$ = …

Note that:

• This series is convergent if |r| < 1.
• divergent if r>1.
• oscillatory if r=1 or r=-1.

Formula: The sum of a geometric series is given by the following formula: a+ar+ar2+ar3+… = $\dfrac{a}{1-r}$, provided that -1<r<1.

Example of a Geometric series:

Find the sum $\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\cdots$.

Answer:

Note that the given series is a geometric series with first term a=1/2 and common ration r= 1/2.

As the value of r lies between -1 and 1, that is, -1<r<1, by the above formula, the sum of the infinite series will be equal to

= $\dfrac{a}{1-r}$

= $\dfrac{\frac{1}{2}}{1-\frac{1}{2}}$ = 1.

So the value of series 1/2 + 1/2² + 1/2³ + … is equal to 1.

Homework: Find the sum $1+\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\cdots$.

p-Series

A series of the form $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n^p}$ (p is a real number) is called a p-series. This series

• converges if p>1.
• diverges if p≤1.

For example, $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n}$ is an example of a p-series with p=1, so it is a divergent series. On the other hand, the p-series $\displaystyle \sum_{n=1}^\infty \dfrac{1}{n^2}$ is convergent as p=2>1.

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