Question:
The sum of the series \(0.2 + 0.004 + 0.00006 + 0.0000008 + \dots\) is
The sum of the series \(0.2 + 0.004 + 0.00006 + 0.0000008 + \dots\) is
Show Hint
Use derivative of GP when n appears in numerator.
Updated On: Apr 15, 2026
- 200/891
- 2000/9801
- 1000/9801
- None of these
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
Write terms as:
\[
0.2 = \frac{2}{10},\quad 0.004=\frac{4}{10^3},\dots
\]
\[
= \sum \frac{2n}{10^{2n-1}}
\]
Step 1: Transform series.
Convert into derivative of GP.
Step 2: Use known result.
\[ \sum_{n=1}^{\infty} \frac{n}{r^n} = \frac{r}{(r-1)^2} \]
Step 3: Final.
\[ = \frac{2000}{9801} \]
Step 1: Transform series.
Convert into derivative of GP.
Step 2: Use known result.
\[ \sum_{n=1}^{\infty} \frac{n}{r^n} = \frac{r}{(r-1)^2} \]
Step 3: Final.
\[ = \frac{2000}{9801} \]
Was this answer helpful?
0
0
Top MET Mathematics Questions
- Let \( f:\mathbb{N} \to \mathbb{N} \) be defined as \[ f(n)= \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] Then \( f \) is:
- MET - 2024
- Mathematics
- types of functions
- Given vectors \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear and \((\vec{a}+\vec{b})\) is collinear with \((\vec{b}+\vec{c})\) which is collinear with \(\vec{a}\), and \(|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}\), find \(|\vec{a}+\vec{b}+\vec{c}|\).
- MET - 2024
- Mathematics
- Addition of Vectors
- Given \(\frac{dy}{dx} + 2y\tan x = \sin x\), \(y=0\) at \(x=\frac{\pi}{3}\). If maximum value of \(y\) is \(1/k\), find \(k\).
- MET - 2024
- Mathematics
- Differential equations
- If \(x = \sin(2\tan^{-1}2)\), \(y = \sin\left(\frac{1}{2}\tan^{-1}\frac{4}{3}\right)\), then:
- MET - 2024
- Mathematics
- Properties of Inverse Trigonometric Functions
- Let \( D = \begin{vmatrix} n & n^2 & n^3 \\ n^2 & n^3 & n^5 \\ 1 & 2 & 3 \end{vmatrix} \). Then \( \lim_{n \to \infty} \frac{M_{11} + C_{33}}{(M_{13})^2} \) is:
- MET - 2024
- Mathematics
- Determinants
View More Questions
Top MET Series Questions
- \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+1} \) is equal to
- \[ \sum_{r=1}^{\infty} \left(3\cdot 2^{-r} - 2\cdot 3^{1-r}\right) \]
- \[ 1 + \frac{3}{1!} + \frac{5}{2!} + \frac{7}{3!} + \cdots \infty \]
- \(\log x + \log x^3 + \log x^5 + \dots + \log x^{2n-1}\) is equal to
- \(\frac{1}{\log_2 10} + \frac{1}{\log_4 10} + \frac{1}{\log_8 10} + \frac{1}{\log_{16} 10}\) is equal to 11
View More Questions
Top MET Questions
- Let \( f:\mathbb{N} \to \mathbb{N} \) be defined as \[ f(n)= \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] Then \( f \) is:
- MET - 2024
- types of functions
- Given vectors \(\vec{a}, \vec{b}, \vec{c}\) are non-collinear and \((\vec{a}+\vec{b})\) is collinear with \((\vec{b}+\vec{c})\) which is collinear with \(\vec{a}\), and \(|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{2}\), find \(|\vec{a}+\vec{b}+\vec{c}|\).
- MET - 2024
- Addition of Vectors
- Given \(\frac{dy}{dx} + 2y\tan x = \sin x\), \(y=0\) at \(x=\frac{\pi}{3}\). If maximum value of \(y\) is \(1/k\), find \(k\).
- MET - 2024
- Differential equations
- Let \( f(x) \) be a polynomial such that \( f(x) + f(1/x) = f(x)f(1/x) \), \( x > 0 \). If \( \int f(x)\,dx = g(x) + c \) and \( g(1) = \frac{4}{3} \), \( f(3) = 10 \), then \( g(3) \) is:
- MET - 2024
- Definite Integral
- A real differentiable function \(f\) satisfies \(f(x)+f(y)+2xy=f(x+y)\). Given \(f''(0)=0\), then \[ \int_0^{\pi/2} f(\sin x)\,dx = \]
- MET - 2024
- Definite Integral
View More Questions