JEE Mains · Maths · STD 11 - 8. sequence and series
The sum of the series \(\sum_{n=1}^{\infty} \frac{n^{2}+6 n+10}{(2 n+1) !}\) is equal to :
- A \(\frac{41}{8} e +\frac{19}{8} e ^{-1}-10\)
- B \(\frac{41}{8} e -\frac{19}{8} e ^{-1}-10\)
- C \(\frac{41}{8} e +\frac{19}{8} e ^{-1}+10\)
- D \(-\frac{41}{8} e +\frac{19}{8} e ^{-1}-10\)
Answer & Solution
Correct Answer
(B) \(\frac{41}{8} e -\frac{19}{8} e ^{-1}-10\)
Step-by-step Solution
Detailed explanation
\(T _{ n }=\frac{ n ^{2}+6 n +10}{(2 n +1) !}=\frac{4 n ^{2}+24 n +40}{4 \cdot(2 n +1) !}\) \(=\frac{(2 n+1)^{2}+20 n+39}{4 \cdot(2 n+1) !}\) \(=\frac{(2 n+1)^{2}+(2 n+1) \cdot 10+29}{4(2 n+1) !}\)…
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
- Let \(m_1\) and \(m_2\) be the slopes of the tangents drawn from the point \(P (4,1)\) to the hyperbola \(H: \frac{y^2}{25}-\frac{x^2}{16}=1\). If \(Q\) is the point from which the tangents drawn to \(H\) have slopes \(\left| m _1\right|\) and \(\left| m _2\right|\) and they make positive intercepts \(\alpha\) and \(\beta\) on the \(x\) axis, then \(\frac{(P Q)^2}{\alpha \beta}\) is equal to \(............\).JEE Mains 2023 Hard
- If the line, \(\frac{{x - 3}}{1} = \frac{{y + 2}}{{ - 1}} = \frac{{z + \lambda }}{{ - 2}}\) lies in the plane, \(2x- 4y + 3z\, = 2\), then the shortest distance between this line and the line, \(\frac{{x - 1}}{{12}} = \frac{y}{9} = \frac{z}{4}\) isJEE Mains 2017 Hard
- Let \(A\) be a matrix of order \(3 \times 3\) and \(|A|=5\). If \(|2 \operatorname{adj}(3 \mathrm{~A} \operatorname{adj}(2 \mathrm{~A}))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma \alpha, \beta, \gamma \in \mathrm{N}\) then \(\alpha+\beta+\gamma\) is equal toJEE Mains 2025 Medium
- If \(f\left( x \right) = {\sin ^{ - 1}}\left( {\frac{{2 \times {3^x}}}{{1 + {9^x}}}} \right)\), then \(f'(-\frac {1}{2})\) equalsJEE Mains 2018 Hard
- Let \(f ( x )\) and \(g ( x )\) be two real polynomials of degree \(2\) and \(1\) respectively. If \(f ( g ( x ))=8 x ^{2}-2 x\), and \(g(f(x))=4 x^{2}+6 x+1\), then the value of \(f(2)+g(2)\) isJEE Mains 2022 Medium
- If the area of the larger portion bounded between the curves \(x^2+y^2=25\) and \(y=|x-1|\) is \(\frac{1}{4}(b \pi+c), b, c \in N\), then \(b+c\) is equal toJEE Mains 2025 Medium
More PYQs from JEE Mains
- The sum of the first ten terms of an A.P. is \(160\) and the sum of the first two terms of a G.P. is \(8\). If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:JEE Mains 2026 Hard
- If \(\int \frac{\cos x d x}{\sin ^{3} x\left(1+\sin ^{6} x\right)^{2 / 3}}=f(x)\left(1+\sin ^{6} x\right)^{1 / \lambda}+c\) where \(c\) is a constant of integration, then \(\lambda f\left(\frac{\pi}{3}\right)\) is equal toJEE Mains 2020 Hard
- If the four complex numbers \(z\), \(\overline{ z }, \overline{ z }-2 \operatorname{Re}(\overline{ z })\) and \(z -2 \operatorname{Re}( z )\) represent the vertices of a square of side \(4\) units in the Argand plane, then \(|z|\) is equal toJEE Mains 2020 Hard
- If \(\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0\,<\,x\,<\,1, a \neq 0\), then the value of \(2 \mathrm{x}^{2}-1\) is :JEE Mains 2021 Hard
- The base of an equilateral triangle is along the line given by \(3x + 4y\,= 9\). If a vertex of the triangle is \((1, 2)\), then the length of a side of the triangle isJEE Mains 2014 Hard
- Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}-\hat{k}, \vec{c}=\lambda \hat{j}+\mu \hat{k}\) and \(\hat{d}\) be a unit vector such that \(\overrightarrow{\mathrm{a}} \times \hat{\mathrm{d}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}} \cdot \hat{\mathrm{d}}=1\), If \(\vec{c}\) is perpendicular to \(\vec{a}\), then \(|3 \lambda \hat{d}+\mu \overrightarrow{\mathrm{c}}|^2\) is equal to _______ .JEE Mains 2025 Medium