How can I find to what this sum converges to?∑n=1∞n(n+1)xn

Letf(x)=x1+x2+x3+⋯Then, for |x|&lt;1f′(x)=1+2x+3x2+4x3+⋯f″(x)=2⋅1⋅x0+3⋅2⋅+4⋅3⋅x2+⋯=∑n=1∞(n+1)nxn−1Soxf″(x)=∑n=1∞(n+1)nxnBut, for |x|&lt;1f(x)=x1−xf′(x)=ddxx1−x=1(1−x)2f″(x)=2(1−x)3So2x(1−x)3=∑n=1∞(n+1)nxnfor |x|&lt;1

"How can I find to what this sum..." was answered by our community

High School
Calculus and Analysis
Calculus 1
Limits and continuity

avalg10o

Answered question

2022-03-24

How can I find to what this sum converges to?

\sum_{n = 1}^{\infty} n (n + 1) x^{n}

Answer & Explanation

Sawyer Anthony

Beginner2022-03-25Added 10 answers

Let

f (x) = x^{1} + x^{2} + x^{3} + \dots

Then, for

| x | < 1

f^{'} (x) = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots

f^{″} (x) = 2 \cdot 1 \cdot x^{0} + 3 \cdot 2 \cdot + 4 \cdot 3 \cdot x^{2} + \dots = \sum_{n = 1}^{\infty} (n + 1) n x^{n - 1}

x f^{″} (x) = \sum_{n = 1}^{\infty} (n + 1) n x^{n}

But, for

| x | < 1

f (x) = \frac{x}{1 - x}

f^{'} (x) = \frac{d}{d x} \frac{x}{1 - x} = \frac{1}{{(1 - x)}^{2}}

f^{″} (x) = \frac{2}{{(1 - x)}^{3}}

\frac{2 x}{{(1 - x)}^{3}} = \sum_{n = 1}^{\infty} (n + 1) n x^{n}

for

| x | < 1

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?