The sum $\displaystyle\sum_{n=1}^{21} \frac{3}{(4 n-1)(4 n+3)}$ is equal to
- $\frac{7}{87}$
- $\frac{7}{29}$
- $\frac{14}{87}$
- $\frac{21}{29}$
The Correct Option is B
Approach Solution - 1
So, The correct option is (B): $\frac7{29}$
Approach Solution -2
\(\overset{21}{\underset{n=1}\sum} \frac3{(4n-1)(4n+3)} = \frac3{4}\overset{21}{\underset{n=1}\sum} \frac1{(4n-1)}-\frac1{(4n+3)}\)
=43n=1∑21(4n−1)(4n+3)(4n+3)−(4n−1)
=43n=1∑214n−11−4n+31
=43(31−71+71−111+111−….+831−871)
=43(31−871)=297
Top Questions on Series
- The value of \[ \lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k + 3)!} \] is:
- In the expansion of $(1 + x)^n$, $\frac{{}^nC_1}{{}^nC_0} + 2 \cdot \frac{{}^nC_2}{{}^nC_1} + 3 \cdot \frac{{}^nC_3}{{}^nC_2} + \dots + n \cdot \frac{{}^nC_n}{{}^nC_{n-1}}$ is equal to:
- If \( \sum_{k=1}^{n} k(k+1)(k-1) = pn^4 + qn^3 + tn^2 + sn \), where \( p, q, t, s \) are constants, then the value of \( s \) is equal to:
- There are four numbers of which the first three are in GP and the last three are in AP, whose common difference is 6. If the first and the last numbers are equal, then the two other numbers are:
- The sum of the infinite series \(1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \dots\) is equal to:
Questions Asked in JEE Main exam
- In YDSE, light of intensity \( 4I \) and \( 9I \) passes through two slits respectively. Difference of maximum and minimum intensity of interference pattern is:
- JEE Main - 2025
- Wave optics
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
- JEE Main - 2025
- Differential Equations
Find the IUPAC name of the compound.
- JEE Main - 2025
- Aromaticity & chemistry of aromatic compounds
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32
- JEE Main - 2025
- Limits and Exponential Functions
- The integral \[ \int_0^\pi \frac{8x}{4\cos^2 x + \sin^2 x} \, dx \text{ is equal to:} \]
- JEE Main - 2025
- Trigonometric Functions
Concepts Used:
Series
A collection of numbers that is presented as the sum of the numbers in a stated order is called a series. As an outcome, every two numbers in a series are separated by the addition (+) sign. The order of the elements in the series really doesn't matters. If a series demonstrates a finite sequence, it is said to be finite, and if it demonstrates an endless sequence, it is said to be infinite.
Read More: Sequence and Series
Types of Series:
The following are the two main types of series are: