10. The sum of the infinite series 1 + 1/2 + 1/3 + 1/4 + 1/5 + … is
a. log 2
b. e
c. π
d. Cannot be determined because the series diverges\
The infinite series 1 + 1/2 + 1/3 + 1/4 + ⋯ is a classic exam question in sequences and series, often asked as “What is the sum of the infinite series 1 + 1/2 + 1/3 + 1/4 + …?”. Understanding why this harmonic series diverges, and why it does not equal log 2, e, or π, is essential for mastering concepts of convergence and divergence in real analysis and competitive exams.
Correct Answer: (d) Cannot be determined because the series diverges
The series 1 + 1/2 + 1/3 + 1/4 + 1/5 + ⋯ is called the harmonic series and it diverges, meaning its partial sums grow beyond all bounds instead of approaching a fixed real number.
Here “cannot be determined” means there is no finite sum; the series diverges to infinity and hence does not equal log 2, e, or π.
Why the Harmonic Series Diverges
To talk about convergence, consider the sequence of partial sums:
Sₙ = 1 + 1/2 + ⋯ + 1/n
A series has a sum only if the limit limₙ→∞ Sₙ exists and is finite.
For the harmonic series, the partial sums grow without bound. One standard proof groups the terms as:
1 + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ⋯
Each group from 1/3 onward has total greater than or equal to 1/2, so the partial sums keep increasing by at least 1/2 again and again, forcing divergence to infinity.
Explanation of Each Option
Option (a) log 2
A famous convergent series is the alternating harmonic series:
1 – 1/2 + 1/3 – 1/4 + ⋯ = log 2
However, the question’s series has all positive terms, so it is not alternating; the harmonic series with all positive terms diverges and therefore cannot equal log 2.
Option (b) e
The number e is approximately 2.71828 and arises as a limit such as limₙ→∞ (1 + 1/n)ⁿ or from the series:
e = ∑ₙ₌₀∞ 1/n!
The harmonic series has terms 1/n, not 1/n!, and its partial sums grow roughly like log n, becoming arbitrarily large and never approaching a finite value like e.
Option (c) π
Many convergent trigonometric and Fourier-type series involve π, such as ∑ 1/n² = π²/6.
But that convergent series is a p-series with exponent p = 2, while the harmonic series corresponds to p = 1, which is exactly the boundary case where the series diverges instead of converging to a finite constant like π.
Option (d) Cannot be determined because the series diverges (Correct)
For convergence, it is necessary that the sequence of partial sums approaches a finite limit.
For the harmonic series, the partial sums are unbounded and increase without limit, so the “sum” of the infinite series is not defined as a real number and cannot be expressed as log 2, e, π, or any other finite constant.
