Series and Sum Calculator with Steps

This calculator will try to find the infinite sum of arithmetic, geometric, power, and binomial series, as well as the partial sum, with steps shown (if possible). It will also check whether the series converges.

Summand:

Variable:

Leave empty for autodetection.

Start value:

End value:

If you need a binomial coefficient $$$C(n,k) = {\binom{n}{k}}$$$, type binomial(n,k).
If you need a factorial $$$n!$$$, type factorial(n).

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find $$$\sum_{n=1}^{\infty} 3^{- n}$$$.

Solution

$$$\sum_{n=1}^{\infty} 3^{- n}$$$ is an infinite geometric series with the first term $$$b=\frac{1}{3}$$$ and the common ratio $$$q=\frac{1}{3}$$$.

By the ratio test, it is convergent.

Its sum is $$$S=\frac{b}{1-q}=\frac{1}{2}$$$.

Therefore,

$${\color{red}{\left(\sum_{n=1}^{\infty} 3^{- n}\right)}}={\color{red}{\left(\frac{1}{2}\right)}}$$

Hence,

$$\sum_{n=1}^{\infty} 3^{- n}=\frac{1}{2}$$

Answer

$$$\sum_{n=1}^{\infty} 3^{- n} = \frac{1}{2} = 0.5$$$A

Series and Sum Calculator with Steps

Calculate series and sums step by step

Your Input

Solution

Answer