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:

如果你需要二項式係數 $$$C(n,k) = {\binom{n}{k}}$$$，請輸入 binomial(n,k)。
如果你需要階乘 $$$n!$$$，請輸入 factorial(n)。

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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

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Solution

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