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:
- x^n/sqrt(n+1)
-
(1000^n)/n!
-
2n
- (x+3)^n/2^n
-
Identical expressions
- (x+ three)^n/ two ^n
- (x plus 3) to the power of n divide by 2 to the power of n
- (x plus three) to the power of n divide by two to the power of n
- (x+3)n/2n
- x+3n/2n
- x+3^n/2^n
- (x+3)^n divide by 2^n
-
Similar expressions
- (x-3)^n/2^n
Sum of series (x+3)^n/2^n
The solution
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[src]
oo ____ \ ` \ n \ (x + 3) ) -------- / n / 2 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(x + 3\right)^{n}}{2^{n}}$$
Sum((x + 3)^n/2^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(x + 3\right)^{n}}{2^{n}}$$
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
$$a_{n} = 2^{- n}$$
and
$$x_{0} = -3$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = -3 + \lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
Let's take the limit
we find
$$R = -1$$
$$\frac{\left(x + 3\right)^{n}}{2^{n}}$$
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
$$a_{n} = 2^{- n}$$
and
$$x_{0} = -3$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = -3 + \lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
Let's take the limit
we find
$$R = -1$$
The answer
[src]
/ 3 x | - + - | 2 2 |3 x| | ------- for |- + -| < 1 | 1 x |2 2| | - - - - | 2 2 < | oo | ___ | \ ` | \ -n n | / 2 *(3 + x) otherwise | /__, \n = 1
$$\begin{cases} \frac{\frac{x}{2} + \frac{3}{2}}{- \frac{x}{2} - \frac{1}{2}} & \text{for}\: \left|{\frac{x}{2} + \frac{3}{2}}\right| < 1 \\\sum_{n=1}^{\infty} 2^{- n} \left(x + 3\right)^{n} & \text{otherwise} \end{cases}$$
Piecewise(((3/2 + x/2)/(-1/2 - x/2), |3/2 + x/2| < 1), (Sum(2^(-n)*(3 + 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