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Sum of series:
1/(3n+1)(3n+4)
9/(9n^2+21n-8)
7^x/(5^x-3)
(sin(2^n))^2/n^2
Identical expressions
nine /(9n^ two +21n- eight)
9 divide by (9n squared plus 21n minus 8)
nine divide by (9n to the power of two plus 21n minus eight)
9/(9n2+21n-8)
9/9n2+21n-8
9/(9n²+21n-8)
9/(9n to the power of 2+21n-8)
9/9n^2+21n-8
9 divide by (9n^2+21n-8)
Similar expressions
9/(9n^2-21n-8)
9/(9n^2+21n+8)

Sum of series/ 9/(9n^2+21n-8)

Sum of series 9/(9n^2+21n-8)

The solution

You have entered [src]

  oo                 
____                 
\   `                
 \           9       
  \   ---------------
  /      2           
 /    9*n  + 21*n - 8
/___,                
n = 1

\sum_{n=1}^{\infty} \frac{9}{\left(9 n^{2} + 21 n\right) - 8}

Sum(9/(9*n^2 + 21*n - 8), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{9}{\left(9 n^{2} + 21 n\right) - 8}

It is a series of species

a_{n} \left(c x - x_{0}\right)^{d n}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}

In this case

a_{n} = \frac{9}{9 n^{2} + 21 n - 8}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\left(21 n + 9 \left(n + 1\right)^{2} + 13\right) \left|{\frac{1}{9 n^{2} + 21 n - 8}}\right|\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

   /         0\                   /        0\              
 3*\-18 + 9*e /*Gamma(14/3)     9*\-1 + 4*e /*Gamma(14/3)  
---------------------------- + ----------------------------
   /          0\                  /          0\            
22*\-20 + 20*e /*Gamma(11/3)   44*\-10 + 10*e /*Gamma(11/3)

\frac{3 \left(-18 + 9 e^{0}\right) \Gamma\left(\frac{14}{3}\right)}{22 \left(-20 + 20 e^{0}\right) \Gamma\left(\frac{11}{3}\right)} + \frac{9 \left(-1 + 4 e^{0}\right) \Gamma\left(\frac{14}{3}\right)}{44 \left(-10 + 10 e^{0}\right) \Gamma\left(\frac{11}{3}\right)}

3*(-18 + 9*exp_polar(0))*gamma(14/3)/(22*(-20 + 20*exp_polar(0))*gamma(11/3)) + 9*(-1 + 4*exp_polar(0))*gamma(14/3)/(44*(-10 + 10*exp_polar(0))*gamma(11/3))

Numerical answer [src]

0.825000000000000000000000000000

0.825000000000000000000000000000

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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How to use it?

Identical expressions

Similar expressions

Sum of series 9/(9n^2+21n-8)

The solution

Examples of finding the sum of a series