To find the value of the double series $\sum_{i=0}^{\infty} \sum_{j=1}^{\infty} 2^{-i} 3^{-j}$, we start by calculating the two inner and outer sums separately. The double series can be rewritten as: \[ \sum_{i=0}^{\infty} \left( \sum_{j=1}^{\infty} 2^{-i} 3^{-j} \right) \]
First, consider the inner sum: \[ \sum_{j=1}^{\infty} 3^{-j} \]
This is a geometric series where the first term \(a=3^{-1}=\frac{1}{3}\) and the common ratio \(r=3^{-1}=\frac{1}{3}\).
The sum of an infinite geometric series is given by \( \frac{a}{1-r} \). Plugging in the values, we have: \[ \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]
Next, substitute this result back into the outer sum: \[ \sum_{i=0}^{\infty} 2^{-i} \cdot \frac{1}{2} \]
This can be expressed as: \[ \frac{1}{2} \sum_{i=0}^{\infty} 2^{-i} \]
Now, consider the series for \( \sum_{i=0}^{\infty} 2^{-i} \), another geometric series with \( a=1 \) and \( r=2^{-1}=\frac{1}{2} \).
The sum is: \[ \frac{1}{1-\frac{1}{2}}=2 \]
Thus, substituting back: \[ \frac{1}{2} \cdot 2 = 1 \]
We find that the value of the double series is \( 1 \), which lies within the given range of 1 to 1.
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? - 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 followingConsider the following infinite series:
$1+r+r^2 + r^3 + ............$ If
$r = 0.3$, then the sum of this infinite series is ________
