QuestionFind the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

Studdy Solution

STEP 1

Assumptions1. We are dealing with an infinite series. . The general term of the series is given by $\frac{n^{10}}{10^{n}}$ .
3. We need to find the sum of this series.

STEP 2

We can't directly calculate the sum of this series because it's an infinite series. However, we can use the power series expansion of a function to find the sum.The power series expansion of a function $f(x)$ is given by $f(x) = \sum_{n=0}^{\infty} f^{(n)}(0) \frac{x^n}{n!}$ where $f^{(n)}(0)$ is the nth derivative of $f(x)$ evaluated at $x=0$ .

STEP 3

We notice that the general term of our series looks like the derivative of $x^n$ evaluated at $x=1$ , divided by $n!$ , and multiplied by $x^n$ .So, we can write the general term of our series as $\frac{n^{10}}{10^{n}} = \frac{1}{n!} \left( \frac{d^{10}}{dx^{10}} x^n \right)_{x=1} \left( \frac{1}{10} \right)^n$

STEP 4

This means our series is the power series expansion of a function $f(x)$ , where $f(x)$ is the10th derivative of $(10x)^n$ evaluated at $x=1/10$ .
We can write this function as $f(x) = \left( \frac{d^{10}}{dx^{10}} (10x)^n \right)_{x=1/10}$

STEP 5

To find the sum of our series, we need to evaluate this function at $x=1/10$ . $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}} = f\left(\frac{1}{10}\right)$

STEP 6

The10th derivative of $(10x)^n$ is $n(n-1)(n-2)...(n-9) \cdot10^n \cdot x^{n-10}$ .

STEP 7

Evaluate this derivative at $x=1/10$ .
$f\left(\frac{1}{10}\right) = \sum_{n=0}^{\infty} n(n-1)(n-2)...(n-9) \cdot10^n \cdot \left(\frac{1}{10}\right)^{n-10}$

STEP 8

implify the expression.
$f\left(\frac{1}{10}\right) = \sum_{n=0}^{\infty} n(n-1)(n-2)...(n-)$

STEP 9

This is the sum of the first $n$ natural numbers raised to the power. This sum is given by the formula $\sum_{n=}^{\infty} n^{} = \frac{}{11}n^{11} + \frac{}{2}n^{} + \frac{5}{6}n^9 - n^7 + \frac{n^6}{2} - \frac{n^5}{2} + \frac{5n^4}{12} - \frac{n^3}{6} + \frac{n^2}{12}$

STEP 10

Substitute $n=\infty$ into the formula to get the sum of the series.
$\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}} = \frac{}{}\infty^{} + \frac{}{2}\infty^{10} + \frac{5}{6}\infty^9 - \infty^7 + \frac{\infty^6}{2} - \frac{\infty^5}{2} + \frac{5\infty^4}{12} - \frac{\infty^3}{6} + \frac{\infty^2}{12}$

STEP 11

This sum is infinite, so the series does not converge.
The sum of the series $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ is $\infty$ .

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