Could someone explain what does it mean by "(and hence $\sum a_n$)"?
Isn't $a_{n+1} \leq a_n$ for all $n$? How does the convergence of $\sum a_{n+1}$ imply $\sum a_n$ converges when we have $\sum a_{n+1} \leq \sum a_n$
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$\begingroup$The two sums differ only by the term $a_1$:
$$\sum_{n=1}^\infty a_n=a_1+\sum_{n=2}^\infty a_n=\sum_{n=1}^\infty a_{n+1}\;.$$
If one of them converges, the other must as well. If you want to be a bit more rigorous about it, look at the sequences of partial sums. If the partial sums of $\sum_{n=1}^\infty a_{n+1}$ are $s_1,s_2,s_3,\dots$, then the partial sums of the original series $\sum_{n=1}^\infty a_n$ are $a_1+s_1,a_1+s_2,a_1+s_3,\dots\;$. If the first sequence of partial sums converges to $L$, then the second converges to $a_1+L$. If the first diverges, so does the second.
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