Q.38 Which of the following statements is true for the series given below?
sn = 1 + 1/√2 + 1/√3 + 1/√4 + … + 1/√n
- sn converges to log(√n)
- sn converges to √n
- sn converges to exp(√n)
- sn diverges
The partial sum sn = 1 + 1/√2 + 1/√3 + ⋯ + 1/√n represents a p-series with p=1/2 < 1. This series diverges as n approaches infinity.
CORRECT ANSWER
sn diverges. The infinite series ∑k=1∞ 1/√k diverges because it fails the integral test: ∫1∞ 1/√x dx = limb→∞ 2√b – 2 = ∞.
Detailed Option Analysis
Option 1: sn converges to log(√n)
Log(√n) simplifies to (1/2)log n, which grows without bound, but this matches the standard harmonic series Hn ≈ log n + γ, not the p=1/2 case where sn ≈ 2√n.
This option confuses it with the divergent harmonic series Hn.
Option 2: sn converges to √n
While sn grows like 2√n + C for some constant C (from Euler-Maclaurin summation), √n alone underestimates the growth.
“Converges to” implies a finite limit, which never occurs. The series still diverges.
Option 3: sn converges to exp(√n)
Exp(√n) grows much faster than any polynomial or root behavior of sn. No series partial sum converges to such a rapidly increasing function.
Option 4: sn diverges
This holds true. Bounds show 2(√n – 1) < sn < 1 + 2√n, both tending to infinity as n → ∞.
Mathematical Proof Summary
Improper Integral: ∫1∞ x-1/2 dx = [2x1/2]1∞ = ∞
p-Series Test: p = 1/2 < 1 → diverges
Asymptotic: sn ∼ 2√n + ζ(1/2) + 1/(2√n) + O(1/n3/2)
