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.

Sum of:

Variable:

Start Value:

If you need `-oo`, type -inf.

End Value:

If you need `oo`, type inf.

If you need a binomial coefficient `C(n,k)=((n),(k))`, type binomial(n,k).

If you need a factorial `n!`, type factorial(n).

Variables in the bounds are assumed to be positive.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Answer

Your input: calculate $$$\sum_{n=1}^{\infty} 3^{- n}$$$

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