1. Convergence

Suppose that $a_k \to \infty$ as $k \to \infty$. Prove that $\displaystyle\sum_{k=1}^{\infty} a_k$ converges if and only if the series $\displaystyle\sum_{k=1}^{\infty} (a_{2k} + a_{2k+1})$ converges.

2. You realize its the same series sans the first term right? So $\sum_{k=1}^N a_k = a_1 + \sum_{k=1}^{(N-1)/2}(a_{2k} + a_{2k+1})$ if N is odd and $\sum_{k=1}^N a_k + a_{N+1} = a_1 + \sum_{k=1}^{N/2}(a_{2k} + a_{2k+1})$
if N is even...

Also, if the terms of the sum are going to infinity, doesn't the series diverge?

$\sum_{k=1}^N a_k = a_1 + \sum_{k=1}^{(N-1)/2}(a_{2k} + a_{2k+1})$ if N is odd and $\sum_{k=1}^N a_k + a_{N+1} = a_1 + \sum_{k=1}^{N/2}(a_{2k} + a_{2k+1})$
if N is even...
I definitely do not understand this. Could anyone explain?

4. Originally Posted by failureequalslearn
i definitely do not understand this. Could anyone explain?
$\sum_{j=1}^{10}j1+2+3+4+5+6+7+8+9+10=1+(2+3)+(3+4) +(5+6)+(7+8)+(9+10)$ $=1+\sum_{j=1}^{\frac{10}{2}}(2j+2j+1)$

5. Originally Posted by FailureEqualsLearn
Suppose that $a_k \to \infty$ as $k \to \infty$. Prove that $\displaystyle\sum_{k=1}^{\infty} a_k$ converges if and only if the series $\displaystyle\sum_{k=1}^{\infty} (a_{2k} + a_{2k+1})$ converges.
Do you all realize that if $\lim _{k \to \infty } (a_k ) = \infty$ then $\sum\limits_k {a_k }$ must diverge.
It cannot converge.

6. Wow I screwed this thread up from the get go. It should read $a_k \to 0$ as $k \to \infty$. My apologies =[.