Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
(-1)^n/2n-1
- (x-1)^n/3^n
-
ln(n+1)/(n+1)
-
sqrt(n)*sin(pi/n^2)
-
Identical expressions
- (x- one)^n/ three ^n
- (x minus 1) to the power of n divide by 3 to the power of n
- (x minus one) to the power of n divide by three to the power of n
- (x-1)n/3n
- x-1n/3n
- x-1^n/3^n
- (x-1)^n divide by 3^n
-
Similar expressions
- (x+1)^n/3^n
Sum of series (x-1)^n/3^n
The solution
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[src]
oo ____ \ ` \ n \ (x - 1) ) -------- / n / 3 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(x - 1\right)^{n}}{3^{n}}$$
Sum((x - 1)^n/3^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(x - 1\right)^{n}}{3^{n}}$$
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
$$a_{n} = 3^{- n}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = 1 + \lim_{n \to \infty}\left(3^{- n} 3^{n + 1}\right)$$
Let's take the limit
we find
$$R = 4$$
$$\frac{\left(x - 1\right)^{n}}{3^{n}}$$
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
$$a_{n} = 3^{- n}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = 1 + \lim_{n \to \infty}\left(3^{- n} 3^{n + 1}\right)$$
Let's take the limit
we find
$$R = 4$$
The answer
[src]
/ 1 x | - - + - | 3 3 | 1 x| | ------- for |- - + -| < 1 | 4 x | 3 3| | - - - | 3 3 < | oo | ___ | \ ` | \ -n n | / 3 *(-1 + x) otherwise | /__, \n = 1
$$\begin{cases} \frac{\frac{x}{3} - \frac{1}{3}}{\frac{4}{3} - \frac{x}{3}} & \text{for}\: \left|{\frac{x}{3} - \frac{1}{3}}\right| < 1 \\\sum_{n=1}^{\infty} 3^{- n} \left(x - 1\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((-1/3 + x/3)/(4/3 - x/3), |-1/3 + x/3| < 1), (Sum(3^(-n)*(-1 + x)^n, (n, 1, oo)), True))
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