Suppose that...

$\displaystyle

\sum\limits_{n = 1}^\infty {a_n }

$ is an infinite series with $\displaystyle

a_n \geqslant 0

$ for alln. Then each partial sum is greater than or equal to its predecessor because $\displaystyle

s_{n + 1} = s_n + a_n

$

$\displaystyle

s_1 \leqslant s_2 \leqslant s_3 \leqslant \cdot \cdot \cdot \leqslant s_n \leqslant s_{n + 1} \leqslant \cdot \cdot \cdot

$

Would anyone mind explaining to me how $\displaystyle

s_{n + 1} = s_n + a_n

$ is true?

Let's say we have: $\displaystyle

\sum\limits_{n = 1}^\infty {n^2 }

$, how about letting n=2.

Let's write out the first couple terms:

$\displaystyle

\sum\limits_{n = 1}^\infty {n^2 } = 1 + 4 + 9 + \cdot \cdot \cdot + n

$

So....

$\displaystyle

\begin{gathered}

s_{n + 1} = s_n + a_n \hfill \\

s_3 = s_2 + a_2 \hfill \\

14 \ne 5 + 4 \hfill \\

\end{gathered}

$

What am I not getting?

Thank you.