Convergence and Divergence of Series — Question 7

*Question 7 Determine whether $\displaystyle\sum_{n=1}^{\infty}\arctan n$ converges or diverges.

1. Evaluate the limit of the summand.

2. Apply the nth-term test and describe the growth of the partial sums.

3. Explain why no comparison or ratio test is needed.

*Question 7 – Solution

The summand has limit $$\lim_{n\to\infty}\arctan n=\frac\pi2\ne0.$$ Because the limit is not zero, the series diverges immediately by the nth-term test.

Since $\arctan n\to\pi/2>1$, the definition of a limit guarantees an $N_0$ such that $\arctan n>1$ for all $n\ge N_0$. Then for $N\ge N_0$, $$s_N\ge s_{N_0-1}+(N-N_0+1),$$ so $s_N\to+\infty$ at least linearly.

Every convergent series must have terms tending to zero: if $s_N\to S$, then $a_N=s_N-s_{N-1}\to S-S=0$. Since this necessary condition fails, comparison, ratio, or integral tests are unnecessary.