The sum of the following infinite series is
$2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots$
Series Analysis: Structure
The given infinite series is: $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots$
The series comprises a constant term and terms derived from powers of prime numbers.
- Constant Term: $2$
- Powers of 2 Series: The terms $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ form a geometric series $\sum_{k=1}^{\infty} \frac{1}{2^k}$.
- Powers of 3 Series: The terms $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$ form a geometric series $\sum_{k=1}^{\infty} \frac{1}{3^k}$.
The total sum (S) is expressed as: $S = 2 + \sum_{k=1}^{\infty} \frac{1}{2^k} + \sum_{k=1}^{\infty} \frac{1}{3^k}$
Geometric Series: Sum Calculation
We use the formula for the sum of an infinite geometric series, $S_{\infty} = \frac{a}{1-r}$, where '$a$' is the first term and '$r$' is the common ratio ($|r| < 1$).
Powers of 2 Series: Sum
For the series $\sum_{k=1}^{\infty} \frac{1}{2^k}$: First term $a = \frac{1}{2}$. Common ratio $r = \frac{1}{2}$. Sum$_2 = \frac{1/2}{1 - 1/2} = \frac{1/2}{1/2} = 1$.
Powers of 3 Series: Sum
For the series $\sum_{k=1}^{\infty} \frac{1}{3^k}$: First term $a = \frac{1}{3}$. Common ratio $r = \frac{1}{3}$. Sum$_3 = \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = \frac{1}{2}$.
Total Series: Sum Calculation
Combine the constant term and the calculated sums:
Total Sum = $2 + \text{Sum}_2 + \text{Sum}_3$ Total Sum = $2 + 1 + \frac{1}{2}$ Total Sum = $3 + \frac{1}{2}$ Total Sum = $\frac{6}{2} + \frac{1}{2} = \frac{7}{2}$
Important Questions from Infinite Series
- Consider the two series, $S_A$ and $S_B$, where
$$S_A = \sum_{n=1}^\infty \frac{n^2}{2^n}$$
$$S_B = 1 + \frac{1}{2} + \frac{1}{8} + \frac{1}{16} + \frac{1}{64} + \frac{1}{128} + \frac{1}{512} + \cdots$$
Which of the following statements is correct for the two given series? - The value of $\sum_{i=0}^{\infty} \sum_{j=1}^{\infty} 2^{-i} 3^{-j}$ is ______________ . (Answer in integer)
- Match each entry of List-1 with a suitable entry in List-2 and choose the correct option.
List-1 List-2 P The sum of the series $\sum_{n=1}^\infty \frac{1}{(n+2)(n+1)}$ is equal to I $\frac{3}{2}$ Q $\lim_{x \to 0} \left( \frac{3}{x^2} \int_0^x \sin(t) dt \right)$ is equal to II $1$ R Let $\frac{a_0}{2} + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx)$ be the Fourier series expansion of the function $f(x) = \frac{1}{2} \sin x - \frac{1}{2} \cos x + \frac{1}{\sqrt{2}} \sin 2x, x \in [0, 2\pi]$. Then, $\sum_{n=0}^\infty (a_n^2 + b_n^2)$ is equal to III $\frac{1}{2}$ Consider the following two series
P: $\sum_{n=1}^{\infty} \frac{1}{n}$
Q: $\sum_{n=1}^{\infty} \frac{1}{n^2}$
Choose the correct option from the followingConsider the following infinite series:
$1+r+r^2 + r^3 + ............$ If
$r = 0.3$, then the sum of this infinite series is ________
