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 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- The series represented by $a_n = \frac{n^2-2n}{3n^2+ n}$ is
- The series $s = \sum_{m=1}^{\infty} \frac{m^2}{3^m} (x - 2)^m$ converges for all $x$ with $|x-2| \le R$ given by
- Which one of the following series is divergent?
The value of the series $1+ \sin x + \cos^2 x + \sin^3 x + \dots$ at $x = \frac{ \pi}{4}$ is __________.
