Series - The Basics — Question 9

Question 9 – Solution

Step 1: Form the finite partial sum.

Write the $N$th partial sum before passing to infinity: $$\begin{aligned} s_N&=(\sqrt2-1)+(\sqrt3-\sqrt2)+\cdots+(\sqrt{N+1}-\sqrt N)\\ &=\sqrt{N+1}-1. \end{aligned}$$

Step 2: Analyze the partial-sum limit.

All interior square-root terms cancel, leaving only the first negative boundary term and final positive boundary term. Because $\sqrt{N+1}-1\to\infty$, the partial sums are unbounded and the series diverges to $+\infty$.

Step 3: Check the nth-term condition.

The summand does approach zero. Rationalization gives $$\sqrt{n+1}-\sqrt n=\frac1{\sqrt{n+1}+\sqrt n}\longrightarrow0.$$

Step 4: Reconcile the two facts.

There is no contradiction: the nth-term condition is necessary, not sufficient. Here every increment is positive and shrinks to zero, but the accumulated total still grows without bound. Since the denominator above is asymptotic to $2\sqrt n$, each increment is asymptotic to $1/(2\sqrt n)$, consistent with a divergent $p$-series scale.