Infinite Series Convergence Analysis
The infinite series is given as $1 + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$.
Let's examine the terms:
- First term: $T_1 = 1$
- Second term: $T_2 = \frac{1}{4}$
- Third term: $T_3 = \frac{1}{8}$
- Fourth term: $T_4 = \frac{1}{16}$
Calculate the ratios between consecutive terms:
- Ratio between $T_2$ and $T_1$: $\frac{T_2}{T_1} = \frac{1/4}{1} = \frac{1}{4}$
- Ratio between $T_3$ and $T_2$: $\frac{T_3}{T_2} = \frac{1/8}{1/4} = \frac{1}{8} \times \frac{4}{1} = \frac{1}{2}$
- Ratio between $T_4$ and $T_3$: $\frac{T_4}{T_3} = \frac{1/16}{1/8} = \frac{1}{16} \times \frac{8}{1} = \frac{1}{2}$
From the second term onwards, the series behaves like a geometric progression: $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$.
This geometric sub-series has:
- First term $a = \frac{1}{4}$
- Common ratio $r = \frac{1}{2}$
Convergence Condition for Geometric Series
A geometric series converges if the absolute value of its common ratio is less than 1, i.e., $|r| < 1$.
For the sub-series, $|r| = |\frac{1}{2}| = \frac{1}{2}$.
Since $\frac{1}{2} < 1$, the geometric sub-series $\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$ converges.
The sum of this convergent sub-series is calculated as $S_{sub} = \frac{a}{1-r} = \frac{1/4}{1 - 1/2} = \frac{1/4}{1/2} = \frac{1}{2}$.
The original series is the sum of the first term and the convergent sub-series: $S = 1 + S_{sub} = 1 + \frac{1}{2} = \frac{3}{2}$.
Because the series sums to a finite value ($\frac{3}{2}$), the series is convergent.
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 $\frac{1}{2}$ The sum of the following infinite series is
$2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \dots$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 following
