Sequence and SeriesMCQMTP June 2023 Series IIQuestion 1860 of 212
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The sum up to infinity of the series S=11⋅2+12⋅3+13⋅4+…\displaystyle S = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots is

Options

A5\displaystyle 5
B1\displaystyle 1
C3/2\displaystyle 3/2
DNone of these
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Correct Answer

✅ Option b — 1\displaystyle 1

All Options:

  • A5\displaystyle 5
  • B1\displaystyle 1
  • C3/2\displaystyle 3/2
  • DNone of these

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Detailed Solution & Explanation

The general term tn\displaystyle t_n of the series can be decomposed using partial fractions:
tn=1n(n+1)=1n−1n+1t_n = \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}

The sum of the first k\displaystyle k terms is:
Sk=∑n=1k(1n−1n+1)S_k = \sum_{n=1}^{k} \left(\frac{1}{n} - \frac{1}{n+1}\right)
Sk=(1−12)+(12−13)+⋯+(1k−1k+1)S_k = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \dots + \left(\frac{1}{k} - \frac{1}{k+1}\right)
Sk=1−1k+1S_k = 1 - \frac{1}{k+1}

Taking the limit as k→∞\displaystyle k \to \infty:
S∞=lim⁡k→∞(1−1k+1)=1S_{\infty} = \lim_{k \to \infty} \left(1 - \frac{1}{k+1}\right) = 1
Hence, **Option B** is the correct answer.

About This Chapter: Sequence and Series

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Arithmetic & Geometric Progressions

This chapter covers Arithmetic Progressions (AP) and Geometric Progressions (GP). Students learn how to find the nth term, sum of n terms, arithmetic/geometric means, and sum to infinity of a GP.

View Official ICAI Syllabus

Exam Strategy Tip

For complex 'sum of series' questions, a great hack is to substitute n = 1 and n = 2 into the question and the options to see which one matches, completely bypassing the formula.

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