Consider the following infinite series: $1+r+r^2 + r^3 + ............$ If $r = 0.3$, then the sum of this infinite series is ________
Infinite Series Sum Calculation
The problem requires finding the sum of the infinite geometric series $1+r+r^2 + r^3 + \dots$ when the common ratio $r$ is $0.3$.
Identify Series Properties
- The series $1+r+r^2 + r^3 + \dots$ is a geometric series.
- The first term ($a$) is $1$.
- The common ratio ($r$) is given as $0.3$.
Check Convergence Condition
An infinite geometric series converges (has a finite sum) only if the absolute value of the common ratio is less than 1, i.e., $|r| < 1$.
- Here, $r = 0.3$.
- Since $|0.3| < 1$, the series converges.
Calculate the Sum
The sum ($S$) of a convergent infinite geometric series is given by the formula:
$S = \frac{a}{1-r}$
Substitute the values $a=1$ and $r=0.3$ into the formula:
$S = \frac{1}{1 - 0.3}$
$S = \frac{1}{0.7}$
$S = \frac{10}{7}$
To compare with the given range, convert the fraction to a decimal:
$S \approx 1.42857...$
Final Result
The calculated sum is approximately $1.42857$. This value falls within the range of 1.4 to 1.43.
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
