Mister Exam

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Sum of series:
log((n+2)/n)^((-1)^(n+1))
a^n*cos(a)+b^n*sin(n*b)
k*(1/3)^k
factorial(6*n)/4^n
Identical expressions
k*(one / three)^k
k multiply by (1 divide by 3) to the power of k
k multiply by (one divide by three) to the power of k
k*(1/3)k
k*1/3k
k(1/3)^k
k(1/3)k
k1/3k
k1/3^k
k*(1 divide by 3)^k

Sum of series/ k*(1/3)^k

Sum of series k*(1/3)^k

The solution

You have entered [src]

-1 + n      
 ___        
 \  `       
  \       -k
  /    k*3  
 /__,       
k = 0

\sum_{k=0}^{n - 1} \left(\frac{1}{3}\right)^{k} k

Sum(k*(1/3)^k, (k, 0, -1 + n))

The rate of convergence of the power series

The answer [src]

       -n        -n
3   3*3     3*n*3  
- - ----- - -------
4     4        2

\frac{3}{4} - \frac{3 \cdot 3^{- n} n}{2} - \frac{3 \cdot 3^{- n}}{4}

3/4 - 3*3^(-n)/4 - 3*n*3^(-n)/2

The graph

Sum of series k*(1/3)^k

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
```