# Infinite Series Calculator

This free online infinite series calculator tool calculates the summation value of a function given a set of limits. With STUDYQUERIES’s online infinite series calculator tool, you can easily calculate values in a fraction of a second.

## What is Infinite Series Calculator?

STUDYQUERIES’ Infinite Series Calculator is an online tool that calculates the summation of infinite series for a given function. Using STUDYQUERIES’s online Infinite Series Calculator, you can quickly calculate the sum of infinite series for a given function.

## How to Use Infinite Series Calculator?

To use the infinite series calculator, follow these steps:

• Step 1: Enter the function in the first input field and enter the summation limits “from” and “to” in the appropriate fields
• Step 2: Click “Submit” to get the results
• Step 3: The summation value will appear in the new window

## What Is Infinite Series?

The sum of infinitely many numbers related in a given way and ordered in a given way. Mathematics and other disciplines such as physics, chemistry, biology, and engineering make use of infinite series.

For an infinite series $$a_1 + a_2 + a_3 +\ldots$$, a quantity $$s_n = a_1 + a_2 +\ldots+ a_n$$, which involves adding only the first $$n$$ terms, is called a partial sum of the series. If $$s_n$$ approaches a fixed number $$S$$ as $$n$$ becomes larger and larger, the series is said to converge. In this case, $$S$$ is called the sum of the series.

Whenever an infinite series does not converge, it is said to diverge. The sum is not assigned a value when there is divergence. For example, the nth partial sum of the infinite series $$1 + 1 + 1 +\ldots$$ is $$n$$. As more terms are added, the partial sum fails to approach any finite value (it grows without bound). As a result, the series diverges. One example of a convergent series is

$$1+\mathbf{\frac{1}{2}}+\mathbf{\frac{1}{4}}+\ldots+\mathbf{\frac{1}{2^n}}$$

As $$n$$ becomes larger, the partial sum approaches $$2$$, which is the sum of this infinite series.

In fact, the series $$1 + r + r^2 + r^3 +\ldots$$ (in the example above $$r$$ equals $$\frac{1}{2}$$ converges to the sum.

$$\mathbf{\frac{1}{(1 − r)}}$$

if $$0 \lt r \lt 1$$ and diverges if $$r \geq 1$$.

The geometric series with ratio r is one of the first infinite series to be studied. Its solution is Zeno of Elea’s paradox regarding the race between Achilles and a tortoise.

The convergence or divergence of a given series can be determined using certain standard tests, but such a determination is not always possible.

In general, if the series $$a_1 + a_2 +\ldots$$ converges, then it must be true that an approaches $$0$$ as $$n$$ becomes larger. Adding or deleting a finite number of terms from a series has no effect on whether or not the series converges.

Moreover, if all the terms in a series are positive, its partial sums will increase, either approaching a finite quantity (convergence) or growing without bound (divergence). Based on this observation, we can perform a comparison test:

• if $$0 ≤ a_n ≤ b_n$$ for all $$n$$ and if $$b_1 + b_2 +\ldots$$ is a convergent infinite series, then $$a_1 + a_2 +\ldots$$ also converges.

The comparison test is reformulated slightly and called the ratio test when applied to a geometric series:

• if $$a_n > 0$$ and if $$\frac{a_{n + 1}}{a_n} \leq r$$ for some $$r \lt 1$$ for every $$n$$, then $$a_1 + a_2 +\ldots$$ converges. For example, the ratio test proves the convergence of the series

$$\mathbf{1+\frac{1}{2}+\frac{1}{3\times2}+\frac{1}{4\times3\times2}+\ldots}$$

There are many mathematical problems that can be solved directly and easily when the function can be expressed as an infinite series involving trigonometric functions (sine and cosine). A rather arbitrary function is broken up into an infinite trigonometric series by applying Fourier analysis or harmonic analysis, and this process is widely used in the study of various wave phenomena.

### Sums and Series

The term infinite series refers to a sum of infinitely many terms written in the form of

$$\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\ldots$$

But what does this mean? We cannot add an infinite number of terms in the same way we can add a finite number of terms. Instead, the value of an infinite series is defined in terms of the limit of partial sums. A partial sum of an infinite series is a finite sum of the form

$$\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\ldots+a_k$$

Let’s take an example for a better understanding

To see how we use partial sums to evaluate infinite series, consider the following example. Suppose oil is seeping into a lake such that $$1000\ gallons$$ enter the lake the first week. During the second week, an additional $$500\ gallons$$ of oil enters the lake. In the third week, $$250\ more\ gallons$$ flow into the lake.

If this pattern continues, each week half as much oil enters the lake as did the previous week. What can we say about the amount of oil in the lake if this goes on forever? Can the amount of oil continue to grow arbitrarily large, or could it approach some finite amount?

To answer this question, we look at the amount of oil in the lake after $$k$$ weeks. Letting $$S_k$$ denote the amount of oil in the lake (measured in thousands of gallons) after $$k$$ weeks, we see that

$$S_1=1$$

$$S_2=1+0.5=1+\frac{1}{2}$$

$$S_3=1+0.5+0.25=1+\frac{1}{2}+\frac{1}{4}$$

$$S_4=1+0.5+0.25+0.125=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}$$

$$S_5=1+0.5+0.25+0.125+0.0625=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}$$

Looking at this pattern, we see that the amount of oil in the lake (in thousands of gallons) after $$k$$ weeks is

$$S_k=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots+\frac{1}{2^{k−1}}=\sum_{n=1}^k\left(\frac{1}{2}\right)^{n−1}$$

We are interested in what happens as $$k\rightarrow \infty$$. Symbolically, the amount of oil in the lake as $$k\rightarrow \infty$$ is given by the infinite series

$$\sum_{n=1}^\infty \left(\frac{1}{2}\right)^{n−1}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\ldots$$

At the same time, as $$k\rightarrow \infty$$, the amount of oil in the lake can be calculated by evaluating $$\lim_{k \to \infty}S_k$$. Therefore, the behavior of the infinite series can be determined by looking at the behavior of the sequence of partial sums $$S_k$$. If the sequence of partial sums $$S_k$$ converges, we say that the infinite series converges, and its sum is given by $$\lim_{k \to \infty}S_k$$. If the sequence $$S_k$$ diverges, we say the infinite series diverges. We now turn our attention to determining the limit of this sequence $$S_k$$.

First, simplifying some of these partial sums, we see that

$$S_1=1$$

$$S_2=1+\frac{1}{2}=\frac{3}{2}$$

$$S_3=1+\frac{1}{2}+\frac{1}{4}=\frac{7}{4}$$

$$S_4=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=\frac{15}{8}$$

$$S_5=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}=\frac{31}{16}$$

Plotting some of these values in Figure, it appears that the sequence $$S_k$$ could be approaching $$2$$.

These data suggest that the sequence $$S_k$$ converges to $$2$$. Later we will provide an analytic argument that can be used to prove that $$\lim_{k \to \infty}S_k=2$$

We rely on the numerical and graphic evidence to convince ourselves that the sequence of partial sums does indeed converge to $$2$$. In this case, since the series of partial sums converges to $$2$$, we say the infinite series converges to $$2$$ and write

$$\sum_{n=1}^\infty \left(\frac{1}{2}\right)^{n−1}=2$$

Returning to the question about the oil in the lake, since this infinite series converges to $$2$$, we conclude that the amount of oil in the lake will get arbitrarily close to $$2000$$ gallons as the amount of time gets sufficiently large.

This series is an example of a geometric series. We discuss geometric series in more detail later in this section.

First, we summarize what it means for an infinite series to converge.

An infinite series is an expression of the form

$$\sum_{n=1}^\infty a_n=a_1+a_2+a_3+\ldots$$

For each positive integer $$k$$, the sum

$$S_k=\sum_{n=1}^ka_n=a_1+a_2+a_3+⋯+a_k$$

is called the kth partial sum of the infinite series. The partial sums form a sequence $$S_k$$. If the sequence of partial sums converges to a real number $$S$$, the infinite series converges. If we can describe the convergence of a series to $$S$$, we call $$S$$ the sum of the series, and we write

$$\sum_{n=1}^\infty a_n=S$$

If the sequence of partial sums diverges, we have the divergence of a series.

Note that the index for a series need not begin with $$n=1$$ but can begin with any value. For example, the series

$$\sum_{n=1}^\infty\left(\frac{1}{2}\right)^{n−1}$$

can also be written as

$$\sum_{n=0}^\infty\left(\frac{1}{2}\right)^n\; \text{or}\; \sum_{n=5}^\infty\left(\frac{1}{2}\right)^{n−5}$$

Often it is convenient for the index to begin at $$1$$, so if for some reason it begins at a different value, we can re-index by making a change of variables. For example, consider the series

$$\sum_{n=2}^\infty \frac{1}{n^2}$$

By introducing the variable $$m=n−1$$, so that $$n=m+1$$, we can rewrite the series as

$$\sum_{m=1}^\infty \frac{1}{(m+1)^2}$$

### The Harmonic Series

A useful series to know about is the harmonic series. The harmonic series is defined as

$$\sum_{n=1}^\infty \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots$$

This series is interesting because it diverges, but it diverges very slowly. By this, we mean that the terms in the sequence of partial sums $$S_k$$ approach infinity, but do so very slowly.

### Algebraic Properties of Convergent Series

Because the sum of a convergent infinite series is a limit of a sequence, the algebraic properties for the series listed below are directly related to the algebraic properties for sequences.

Let $$\sum_{n=1}^\infty a_n$$ and $$\sum_{n=1}^\infty b_n$$ be convergent series. Then the following algebraic properties hold.

• The series $$\sum_{n=1}^\infty(a_n+b_n)$$ converges, and $$\sum^\infty_{n=1}(a_n+b_n)=\sum^\infty_{n=1}a_n+\sum^\infty_{n=1}b_n$$. (Sum Rule)
• The series $$\sum_{n=1}^\infty(a_n−b_n)$$ converges, and $$\sum^\infty_{n=1}(a_n−b_n)=\sum^\infty_{n=1}a_n−\sum^\infty_{n=1}b_n$$. (Difference Rule)
• For any real number $$c$$, the series $$\sum_{n=1}^\infty ca_n$$ converges, and $$\sum^\infty_{n=1}ca_n=c\sum^\infty_{n=1}a_n$$. (Constant Multiple Rule)

### Geometric Series

A geometric series is any series that we can write in the form

$$a+ar+ar^2+ar^3+\dots=\sum_{n=1}^\infty ar^{n−1}$$

Because the ratio of each term in this series to the previous term is $$r$$, the number $$r$$ is called the ratio. We refer to $$a$$ as the initial term because it is the first term in the series. For example, the series

$$\sum_{n=1}^\infty \left(\frac{1}{2}\right)^{n−1}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots$$

is a geometric series with initial term $$a=1$$ and ratio $$r=\frac{1}{2}$$.

In general, when does a geometric series converge? Consider the geometric series

$$\sum_{n=1}^\infty ar^{n−1}$$

when $$a\gt 0$$. Its sequence of partial sums $$S_k$$ is given by

$$S_k=\sum_{n=1}^kar^{n−1}=a+ar+ar^2+\ldots+ar^{k−1}$$

Consider the case when $$r=1$$. In that case,

$$S_k=a+a(1)+a(1)^2+\ldots+a(1)^{k−1}=ak$$

Since $$a\gt 0$$, we know $$ak\rightarrow \infty$$ as $$k\rightarrow \infty$$. Therefore, the sequence of partial sums is unbounded and thus diverges.

Consequently, the infinite series diverges for $$r=1$$. For $$r\neq 1$$, to find the limit of $$S_k$$, multiply Equation by $$1−r$$. By doing so, we see that

$$(1−r)S_k=a(1−r)(1+r+r^2+r^3+\ldots+r^{k−1})\\ =a[(1+r+r^2+r^3+\ldots+r^{k−1})−(r+r^2+r^3+\ldots+r^k)]=a(1−r^k)$$

All the other terms cancel out. Therefore,

$$S_k=\frac{a(1−r^k)}{1−r}$$ for $$r\neq 1$$.

From our discussion in the previous section, we know that the geometric sequence $$rk\rightarrow 0$$ if $$|r|\lt 1$$ and that $$rk$$ diverges if $$|r|\gt 1$$ or $$r=\pm 1$$. Therefore, for $$|r|\lt 1$$,$$S_k\rightarrow \frac{a}{1−r}$$ and we have

$$\sum_{n=1}^\infty ar^{n−1}=\frac{a}{1−r}\; \text{if}\; |r|<1$$

If $$|r|≥1$$,$$S_k$$ diverges, and therefore

$$\sum_{n=1}^\infty ar^{n−1} \; \text{diverges if}\; |r|≥1$$

## FAQs

How do you find infinite series?

In finding the sum of the given infinite geometric series If r<1 is then the sum is given as Sum = a/(1-r). In this infinite series formula, a = first term of the series and r = common ratio between two consecutive terms and −1<r<1.

What is the formula of infinite terms?

The infinite geometric series formula is S∞ = a/(1 – r), where a is the first term and r is the common ratio.

What is an infinite series example?

When we have an infinite sequence of values: 12, 14, 18, 116, … we get an infinite series. “Series” sounds like it is the list of numbers, but it is actually when we add them together.

What is the formula for finding the sum of a series?

To find the sum of an arithmetic sequence, use the formula Sn=n(a1+an)2 where Sn is the sum of n terms, a1 is the first term in the sequence, and an is the nth term.