Answer
Convergent with sum $1$
Work Step by Step
We are given the series:
$\sum_{n=1}^{\infty}\left(\dfrac{1}{\sqrt n}-\dfrac{1}{\sqrt{n+1}}\right)$.
A partial sum for m terms can be written:
$S_m=\sum_{n=1}^{m}\left(\dfrac{1}{\sqrt n}-\dfrac{1}{\sqrt{n+1}}\right)$=
$=\dfrac{1}{\sqrt{1}}-\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{3}}+....+\dfrac{1}{\sqrt{m}}-\dfrac{1}{\sqrt{m+1}}$
$=1-\dfrac{1}{\sqrt{m+1}}$
Determine at least 10 partial sums:
$S_1=1-\dfrac{1}{\sqrt{2}}\approx 0.29289$
$S_2=1-\dfrac{1}{\sqrt{3}}\approx 0.42265$
$S_3=1-\dfrac{1}{\sqrt{4}}=0.5$
$S_4=1-\dfrac{1}{\sqrt{5}}\approx 0.55279$
$S_5=1-\dfrac{1}{\sqrt{6}}\approx 0.59175$
$S_{10}=1-\dfrac{1}{\sqrt{11}}\approx 0.69849$
$S_{25}=1-\dfrac{1}{\sqrt{26}}\approx 0.80388$
$S_{50}=1-\dfrac{1}{\sqrt{51}}\approx 0.85997$
$S_{100}=1-\dfrac{1}{\sqrt{101}}\approx 0.90050$
$S_{500}=1-\dfrac{1}{\sqrt{501}}\approx 0.95532$
$S_{1000}=1-\dfrac{1}{\sqrt{1001}}\approx 0.96839$
Graph both the sequence of terms (in blue color) and the sequence of partial sums (in red color).
When $m\rightarrow \infty$, $\dfrac{1}{\sqrt{m+1}}\rightarrow 0$, so $S_m\rightarrow 1-0=1$, therefore $a_n\rightarrow 1$. Therefore the series is convergent, its sum being 1.
