TS EAMCET · Maths · Sequences and Series
\(\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\frac{1}{4 \cdot 9}+\ldots\) is equal to
- A \(2 \log _e 2-2\)
- B \(2-\log _e 2\)
- C \(2 \log _e 4\)
- D \(\log _e 4\)
Answer & Solution
Correct Answer
(B) \(2-\log _e 2\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let } \quad S=\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 7}+\frac{1}{4 \cdot 9}+\ldots \\ & \therefore \quad T_n=\frac{1}{n(2 n+1)} \\ & =\frac{1}{n}-\frac{2}{(2 n+1)} \\ & \Rightarrow \quad S=\sum_{n=1}^{\infty}…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If the position vectors of the points \(A, B, C, D\) given by \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\), \(\frac{1}{4}(7 \hat{\mathbf{i}}+15 \hat{\mathbf{j}}+15 \hat{\mathbf{k}}) \text { and } \frac{1}{3}[7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+(5+3 a) \hat{\mathbf{k}}]\) respectively are such that \(|\mathbf{A C}|=|\mathbf{B D}|\), then \(16(3 a-1)^2=\)TS EAMCET 2019 Medium
- If the combined equation of the diagonals of the square formed by the pairs of lines \(x y+4 x-5 y-20=0\) and \(x y-5 x+4 y-20=0\) is \(x^2-y^2-k x+l y=0\), then \(k+l=\).TS EAMCET 2019 Medium
- Let \(\mathbb{N}\) be the set of positive integers. The number of distinct triplets \((x, y, z)\) satisfying \(x, y, z \in \mathbb{N}, x < y < z\) and \(x+y+z=12\) isTS EAMCET 2022 Easy
- The perpendicular distance from the origin to the plane containing the points having position vectors isTS EAMCET 2021 Easy
- \(\sin ^{-1} \frac{4}{5}+2 \tan ^{-1} \frac{1}{3}\) is equal toTS EAMCET 2005 Medium
- If \(\omega\) is a complex cube root of unity, then \(\sum_{x=1}^{10}\left((\omega x+2)\left(\omega^2 x+2\right)-3\right)\)TS EAMCET 2020 Easy
More PYQs from TS EAMCET
- \(\mathrm{O}(0,0,0), \mathrm{A}(3,1,4), \mathrm{B}(1,3,2)\) and \(\mathrm{C}(0,4,-2)\) are the vertices of a tetrahedron. If G is the centroid of the tetrahedron and \(\mathrm{G}_1\) is the centroid of its face ABC, then the point which divides \(\mathrm{GG}_1\) in the ratio \(1: 2\) isTS EAMCET 2025 Medium
- In hydrogen atom, the minimum energy required to excite an electron from orbit to the orbit isTS EAMCET 2022 Medium
- In the preparation of chlorine by the electrolysis of brine, the reaction taking place at the anode isTS EAMCET 2019 Medium
- Heating a mixture of \(\mathrm{Cu}_2 \mathrm{O}\) and \(\mathrm{Cu}_2 \mathrm{~S}\) will giveTS EAMCET 2017 Medium
- The correct option for the first ionisation enthalpy (in \(\mathrm{kJ} \mathrm{mol}^{-1}\) ) of \(\mathrm{Li}, \mathrm{Na}, \mathrm{K}\) and \(\mathrm{Cs}\) respectively isTS EAMCET 2018 Easy
- Let \(f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2, & x \leq 1 \ x+\log _2\left(b^2+7\right), & x>1\end{array}\right.\). Then the set of all possible values of \(b\) such that \(f(1)\) is the maximum value of \(f(x)\) isTS EAMCET 2022 Medium