Series

115. Series

Given the series $(𝑎_{𝑛}) = 𝑎_{1} + 𝑎_{2} + \dots$ , then

\sum_{𝑛 = 1}^{\infty} 𝑎_{𝑛} = 𝑎_{1} + 𝑎_{2} + \dots

We form a sequence associated with the series, the sequence $(𝑠_{𝑛})$ of its partial sums:

\begin{aligned} 𝑠_{1} & = 𝑎_{1} \\ 𝑠_{2} & = 𝑎_{1} + 𝑎_{2} \\ 𝑠_{3} & = 𝑎_{1} + 𝑎_{2} + 𝑎_{3} \\ ⋮ \\ 𝑠_{𝑛} & = 𝑎_{1} + 𝑎_{2} + \dots + 𝑎_{𝑛} \\ ⋮ \end{aligned}

Example

\sum_{𝑛 = 1}^{\infty} \frac{1}{𝑛^{2}}

Example

\sum_{𝑛 = 1}^{\infty} \frac{{(- 1)}^{𝑛 + 1}}{𝑛}

Example

\sum_{𝑛 = 1}^{\infty} \frac{1}{𝑛}