Mister Exam

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Sum of series:
sqrt((n+4)/((n^4)+4))
factorial(n+2)/n^n
ln((n+2)/n)
6/((3*n-1)*(3*n+5))
Identical expressions
sqrt((n+ four)/((n^ four)+ four))
square root of ((n plus 4) divide by ((n to the power of 4) plus 4))
square root of ((n plus four) divide by ((n to the power of four) plus four))
√((n+4)/((n^4)+4))
sqrt((n+4)/((n4)+4))
sqrtn+4/n4+4
sqrt((n+4)/((n⁴)+4))
sqrtn+4/n^4+4
sqrt((n+4) divide by ((n^4)+4))
Similar expressions
sqrt((n+4)/((n^4)-4))
sqrt((n-4)/((n^4)+4))

Sum of series/ sqrt((n+4)/((n^4)+4))

Sum of series sqrt((n+4)/((n^4)+4))

The solution

You have entered [src]

  oo               
____               
\   `              
 \         ________
  \       / n + 4  
   )     /  ------ 
  /     /    4     
 /    \/    n  + 4 
/___,              
n = 1

\sum_{n=1}^{\infty} \sqrt{\frac{n + 4}{n^{4} + 4}}

Sum(sqrt((n + 4)/(n^4 + 4)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\sqrt{\frac{n + 4}{n^{4} + 4}}

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} = \sqrt{\frac{n + 4}{n^{4} + 4}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 4} \sqrt{\left(n + 1\right)^{4} + 4}}{\sqrt{n + 5} \sqrt{n^{4} + 4}}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

  oo              
_____             
\    `            
 \        _______ 
  \     \/ 4 + n  
   \   -----------
   /      ________
  /      /      4 
 /     \/  4 + n  
/____,            
n = 1

\sum_{n=1}^{\infty} \frac{\sqrt{n + 4}}{\sqrt{n^{4} + 4}}

Sum(sqrt(4 + n)/sqrt(4 + n^4), (n, 1, oo))

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 sqrt((n+4)/((n^4)+4))

The solution

Examples of finding the sum of a series