Evaluate the Summation sum from n=1 to infinity of 2^n
The question asks for the evaluation of an infinite sum, or series, where each term in the sum is an exponential function of the form 2 raised to the power of n, with n starting at 1 and continuing without end. In mathematical notation, this is represented as the sum of 2^n as n goes from 1 to infinity. The task is to determine the total value of this sum if it converges, meaning it approaches a specific number as more and more terms are added.
$\sum_{n = 1}^{\infty} 2^{n}$
Answer
Solution:
Step 1:
To determine the sum of an infinite geometric series, we use the formula $S = \frac{a}{1 - r}$, where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio.
Step 2:
Calculate the common ratio $r$ by using the formula $r = \frac{a_{n+1}}{a_n}$.
Step 2.1:
Insert the terms $a_n$ and $a_{n+1}$ into the formula to find $r$. $r = \frac{2^{n+1}}{2^n}$
Step 2.2:
Eliminate the same factors in $2^{n+1}$ and $2^n$.
Step 2.2.1:
Extract $2^n$ from $2^{n+1}$. $r = \frac{2^n \cdot 2}{2^n}$
Step 2.2.2:
Remove the identical factors.
Step 2.2.2.1:
Apply multiplication by $1$. $r = \frac{2^n \cdot 2}{2^n \cdot 1}$
Step 2.2.2.2:
Cross out the common factor. $r = \frac{\cancel{2^n} \cdot 2}{\cancel{2^n} \cdot 1}$
Step 2.2.2.3:
Simplify the expression. $r = \frac{2}{1}$
Step 2.2.2.4:
Divide $2$ by $1$. $r = 2$
Step 3:
Determine if the series converges. Since $|r| \geq 1$, the series does not converge.
Knowledge Notes:
Infinite geometric series are sequences of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The sum of an infinite geometric series can be found if $|r| < 1$ using the formula $S = \frac{a}{1 - r}$, where $a$ is the first term of the series. If $|r| \geq 1$, the series does not have a finite sum and is considered to diverge.
In this problem, the series is $2^n$ for $n$ starting from 1 to infinity. This is a geometric series where each term is twice the previous term, indicating that the common ratio $r$ is 2. Since the common ratio is greater than 1, the series diverges, meaning it does not sum to a finite value.
