1. ## Nondecreasing Partial Sums

Suppose that...

$
\sum\limits_{n = 1}^\infty {a_n }
$
is an infinite series with $
a_n \geqslant 0
$
for all n. Then each partial sum is greater than or equal to its predecessor because $
s_{n + 1} = s_n + a_n
$

$
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 $
s_{n + 1} = s_n + a_n
$
is true?

Let's say we have: $
\sum\limits_{n = 1}^\infty {n^2 }
$

Let's write out the first couple terms:

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

So....

$
\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.

2. Originally Posted by RedBarchetta
Suppose that...

$
\sum\limits_{n = 1}^\infty {a_n }
$
is an infinite series with $
a_n \geqslant 0
$
for all n. Then each partial sum is greater than or equal to its predecessor because $
s_{n + 1} = s_n + a_n
$
That's not correct. It should be $s_{n + 1} = s_n + a_{n+1}$ ( $s_n$ is the sum of the first n terms; you need to add the (n+1)th term to that in order to get the sum of the first (n+1) terms).

3. Thank you. Just a misprint in the book.