Mister Exam

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Sum of series:
(7^n+3^n)/21^n
log(1+1/n)-log(1+1/(n+1))
38
sen(na)
Identical expressions
nine /(9n^ two -21n- eight)
9 divide by (9n squared minus 21n minus 8)
nine divide by (9n to the power of two minus 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 = 0

\sum_{n=0}^{\infty} \frac{9}{\left(9 n^{2} - 21 n\right) - 8}

Sum(9/(9*n^2 - 21*n - 8), (n, 0, 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(9 \left|{\frac{\frac{7 n}{3} - \left(n + 1\right)^{2} + \frac{29}{9}}{- 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\           
  9*\-32 + 8*e /*Gamma(4/3)   3*\27 + 9*e /*Gamma(4/3)
- ------------------------- - ------------------------
     /        0\                /        0\           
  16*\-5 + 5*e /*Gamma(1/3)   8*\-5 + 5*e /*Gamma(1/3)

- \frac{3 \left(9 e^{0} + 27\right) \Gamma\left(\frac{4}{3}\right)}{8 \left(-5 + 5 e^{0}\right) \Gamma\left(\frac{1}{3}\right)} - \frac{9 \left(-32 + 8 e^{0}\right) \Gamma\left(\frac{4}{3}\right)}{16 \left(-5 + 5 e^{0}\right) \Gamma\left(\frac{1}{3}\right)}

-9*(-32 + 8*exp_polar(0))*gamma(4/3)/(16*(-5 + 5*exp_polar(0))*gamma(1/3)) - 3*(27 + 9*exp_polar(0))*gamma(4/3)/(8*(-5 + 5*exp_polar(0))*gamma(1/3))

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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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series