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:
-
sqrt(1+5n/n)
-
1/(n((lnn)^2))
-
(5/2^n+1)+ln(n/n+1)
-
3^(-n)
-
Identical expressions
- sqrt(one +5n/n)
- square root of (1 plus 5n divide by n)
- square root of (one plus 5n divide by n)
- √(1+5n/n)
- sqrt1+5n/n
- sqrt(1+5n divide by n)
-
Similar expressions
- sqrt((1+5*n)/n)
- sqrt(1-5n/n)
Sum of series sqrt(1+5n/n)
The solution
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oo ____ \ ` \ _________ \ / 5*n / / 1 + --- / \/ n /___, n = 1
$$\sum_{n=1}^{\infty} \sqrt{1 + \frac{5 n}{n}}$$
Sum(sqrt(1 + (5*n)/n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sqrt{1 + \frac{5 n}{n}}$$
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
$$a_{n} = \sqrt{6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
$$\sqrt{1 + \frac{5 n}{n}}$$
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
$$a_{n} = \sqrt{6}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
Numerical answer
The series diverges
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