True or False? #2

A. (counterexample)

$a_n=\sqrt{n+1}-\sqrt{n}=\dfrac{1}{\sqrt{n+1}+\sqrt{n}}$

$\displaystyle \lim_{n\to\infty} a_n=0,$ but $\displaystyle \lim_{n\to\infty} \sum_{k=1}^n a_k$ doesn't have a value.

$\therefore~\boxed{\text{FALSE}}$

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B.

If $\displaystyle \lim_{n\to\infty} \sum_{k=1}^n a_k$ has a value, then $\displaystyle \lim_{n\to\infty} \sum_{k=1}^{n+1} a_k$ also has the same value.

Therefore, $\displaystyle \lim_{n\to\infty} \sum_{k=1}^{n+1} a_k-\lim_{n\to\infty} \sum_{k=1}^n a_k=\lim_{n\to\infty}a_n=0.$

$\therefore~\boxed{\text{TRUE}}$

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C. (counterexmple)

$a_n=(-1)^n.$

$\displaystyle \lim_{n\to\infty} a_{2n}=1,$ but $\displaystyle \lim_{n\to\infty} a_{n}$ doesn't have a value.

$\therefore~\boxed{\text{FALSE}}$

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From above, only B is true.