Question

Which of the following is a convergent series?

• A. $\Sigma_{n=1}^{\infty}(-1)^{n}\frac{1}{n}$
• B. $\Sigma_{n=1}^{\infty}a_{n}$ where $a_{n}=2, \forall n\in N$
• C. $\Sigma_{n=1}^{\infty}\frac{1}{n^{\frac{3}{4}}}$
• D. $\Sigma_{n=1}^{\infty}3^{n}$

Hint:

Remember the p-series rule: $\Sigma \frac{1}{n^p}$ converges only if $p > 1$.

Answer 1

The Correct Option is A

Step 1: Concept A series converges if the sum of its terms approaches a finite limit.

Step 2: Meaning Option (A) is the alternating harmonic series. According to Leibniz's test, an alternating series converges if the terms decrease monotonically to zero.

Step 3: Analysis Option (B) diverges because the terms do not approach zero. Option (C) is a p-series with $p = 3/4$, which diverges since $p \le 1$. Option (D) is a geometric series with $r=3$, which diverges since $|r| \ge 1$.

Step 4: Conclusion Only the alternating harmonic series in (A) satisfies the criteria for convergence. Final Answer: (A)