Sequences & SerieseasyFree

Leibniz Series Regrouped: Sum 1/(1·3)+1/(5·7)+... | JEE

JEE Maths question with a full step-by-step solution.

Question

1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\cdots=\frac{\pi}{4}

, then the value of

\frac{1}{1\cdot3}+\frac{1}{5\cdot7}+\frac{1}{9\cdot11}+\cdots

\frac{\pi}{8}

correct

\frac{\pi}{6}

\frac{\pi}{4}

\frac{\pi}{36}

Solution

Step 1: Pair the given series:

\frac{\pi}{4}=\left(1-\frac13\right)+\left(\frac15-\frac17\right)+\left(\frac19-\frac{1}{11}\right)+\cdots.

Step 2: Each pair is twice a required term, since

1-\frac13=\frac{2}{1\cdot3}

\frac15-\frac17=\frac{2}{5\cdot7}

\ldots

\frac{\pi}{4}=2\left(\frac{1}{1\cdot3}+\frac{1}{5\cdot7}+\frac{1}{9\cdot11}+\cdots\right).

Step 3: So the required sum is

\dfrac{\pi}{8}.

Correct answer: (1)

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.