Mister Exam

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Sum of series:
1/(n(n+1))
(3^(n+3))/(4^(n-1)*5^n)
1/4n^2+1
(2n+6)*tg(15/(9n+11))
Identical expressions
(2n+ six)*tg(fifteen /(9n+ eleven))
(2n plus 6) multiply by tg(15 divide by (9n plus 11))
(2n plus six) multiply by tg(fifteen divide by (9n plus eleven))
(2n+6)tg(15/(9n+11))
2n+6tg15/9n+11
(2n+6)*tg(15 divide by (9n+11))
Similar expressions
(2n-6)*tg(15/(9n+11))
(2n+6)*tg(15/(9n-11))

Sum of series/ (2n+6)*tg(15/(9n+11))

Sum of series (2n+6)*tg(15/(9n+11))

The solution

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  oo                         
 ___                         
 \  `                        
  \                /   15   \
   )  (2*n + 6)*tan|--------|
  /                \9*n + 11/
 /__,                        
n = 1

$$\sum_{n=1}^{\infty} \left(2 n + 6\right) \tan{\left(\frac{15}{9 n + 11} \right)}$$

Sum((2*n + 6)*tan(15/(9*n + 11)), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\left(2 n + 6\right) \tan{\left(\frac{15}{9 n + 11} \right)}$$
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
$$a_{n} = \left(2 n + 6\right) \tan{\left(\frac{15}{9 n + 11} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 6\right) \left|{\frac{\tan{\left(\frac{15}{9 n + 11} \right)}}{\tan{\left(\frac{15}{9 n + 20} \right)}}}\right|}{2 n + 8}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

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 (2n+6)*tg(15/(9n+11))

The solution

Examples of finding the sum of a series