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:
-
n-1
- (2^n+a)/(b*(3^n))
-
(-1)^n*3^n/factorial(n)
- xn^n/2^n
-
Identical expressions
- three - one /(x*i- one)
- 3 minus 1 divide by (x multiply by i minus 1)
- three minus one divide by (x multiply by i minus one)
- 3-1/(xi-1)
- 3-1/xi-1
- 3-1 divide by (x*i-1)
-
Similar expressions
- 3-1/xi-1
- 3-1/(x*i+1)
- 3+1/(x*i-1)
Sum of series 3-1/(x*i-1)
The solution
You have entered
[src]
oo ___ \ ` \ / 1 \ ) |3 - -------| / \ x*i - 1/ /__, i = 1
$$\sum_{i=1}^{\infty} \left(3 - \frac{1}{i x - 1}\right)$$
Sum(3 - 1/(x*i - 1), (i, 1, oo))
The radius of convergence of the power series
Given number:
$$3 - \frac{1}{i x - 1}$$
It is a series of species
$$a_{i} \left(c x - x_{0}\right)^{d i}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}$$
In this case
$$a_{i} = 3 - \frac{1}{i x - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty} \left|{\frac{3 - \frac{1}{i x - 1}}{3 - \frac{1}{x \left(i + 1\right) - 1}}}\right|$$
Let's take the limit
we find
$$3 - \frac{1}{i x - 1}$$
It is a series of species
$$a_{i} \left(c x - x_{0}\right)^{d i}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}$$
In this case
$$a_{i} = 3 - \frac{1}{i x - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty} \left|{\frac{3 - \frac{1}{i x - 1}}{3 - \frac{1}{x \left(i + 1\right) - 1}}}\right|$$
Let's take the limit
we find
True
False
The answer
[src]
// / 1\ \
|| zoo*Gamma|2 - -| 2 |
|| \ x/ im (x) (-1 + re(x))*re(x) (-1 + 2*re(x))*re(x) |
||--------------------- for 1 - --------------- + ------------------ - -------------------- < 0|
|| / 1\ 2 2 2 2 2 2 |
||(-1 + x)*Gamma|1 - -| im (x) + re (x) im (x) + re (x) im (x) + re (x) |
|| \ x/ |
|| |
oo - |< oo |
|| ___ |
|| \ ` |
|| \ 1 |
|| ) -------- otherwise |
|| / -1 + i*x |
|| /__, |
|| i = 1 |
\\ /
$$- \begin{cases} \frac{\tilde{\infty} \Gamma\left(2 - \frac{1}{x}\right)}{\left(x - 1\right) \Gamma\left(1 - \frac{1}{x}\right)} & \text{for}\: \frac{\left(\operatorname{re}{\left(x\right)} - 1\right) \operatorname{re}{\left(x\right)}}{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} - \frac{\left(2 \operatorname{re}{\left(x\right)} - 1\right) \operatorname{re}{\left(x\right)}}{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} + 1 - \frac{\left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2}} < 0 \\\sum_{i=1}^{\infty} \frac{1}{i x - 1} & \text{otherwise} \end{cases} + \infty$$
oo - Piecewise((±oo*gamma(2 - 1/x)/((-1 + x)*gamma(1 - 1/x)), 1 - im(x)^2/(im(x)^2 + re(x)^2) + (-1 + re(x))*re(x)/(im(x)^2 + re(x)^2) - (-1 + 2*re(x))*re(x)/(im(x)^2 + re(x)^2) < 0), (Sum(1/(-1 + i*x), (i, 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