Question

convergence test

Answer 1

\[\sum (-1)^{n} \sqrt{n+1}-\sqrt{n}\]

Answer 2

I know lim=0, but I cannot tell if it decreases or not since it still goes to infinity wouldn't it be divergent?

Answer 3

if n is even then:
\[{a_n} = \sqrt {n + 1} - \sqrt n > 0\]

Answer 4

whereas if n is odd:
\[{a_n} = - \sqrt {n + 1} - \sqrt n < 0\]

Answer 5

I still dont understand

Answer 6

your series is not a series with positive terms

Answer 7

so we have to distinguish those two cases, namely n even and n odd

Answer 8

ok

Answer 9

now, if n is even we can write:
\[\Large \begin{gathered}
{a_n} = \sqrt {n + 1} - \sqrt n = \frac{{n + 1 - n}}{{\sqrt {n + 1} + \sqrt n }} = \hfill \\
\hfill \\
= \frac{1}{{\sqrt {n + 1} + \sqrt n }} \geqslant \frac{1}{{2\sqrt {n + 1} }} \hfill \\
\end{gathered} \]

Answer 10

please tell me when I may continue

Answer 11

ok

Answer 12

since the series
\[\Large \sum {\frac{1}{{2\sqrt {n + 1} }}} \]
is a divergent series, then we can conclude that our original series is divergent if n is even

Answer 13

ok

Answer 14

now we have to consider the case n odd

Answer 15

here we can write:
\[\Large \sum {\left( { - \sqrt {n + 1} - \sqrt n } \right)} = - \sum {\left( {\sqrt {n + 1} + \sqrt n } \right)} \]