Show that if p and q are positive integers, then the radius of convergence of the power series below is R= +∞
\(\sum_{\mathrm{k}=2}^{+\infty} \frac{(\mathrm{k}+\mathrm{p}) !}{\mathrm{k} !(\mathrm{k}+\mathrm{q}) !} \mathrm{x}^{\mathrm{k}}\)
series converges for all values of x.
Therefore radius of convergence of series is R=+∞
Ratio test:
=>series converges for all values of x
Therefore radius of convergence of series is R=+∞
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Show that if p and q are positive integers, then the radius of convergence of the power series below is R= +∞
9. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.)a) \(\sum_{n=0}^{\infty}\left(\frac{3 x}{5}\right)^{n}\)b) \(\sum_{n=0}^{\infty} \frac{2^{n}(x-2)^{n}}{3 n}\)
17 Use a power series to approximate the definite integral, \(I\), to six decimal places.$$ \int_{0}^{0.4} \frac{x^{5}}{1+x^{7}} d x $$Find the radius of convergence, \(R\), of the series.$$ \sum_{n=1}^{\infty} \frac{x^{n+4}}{4 n !} $$$$ R= $$Find the interval, \(I\), of convergence of the series. (Enter your answer using interval notation.) \(I=\)
7. Use the Alternating Series Test to determine the convergence or divergence of the series a) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} \sqrt{n}}{2 n+1}\)b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n} n}{2 n-1}\)8. Use the Ratio Test or the Root Test to determine the convergence or divergence of the seriesa) \(\sum_{n=0}^{\infty}\left(\frac{4 n-1}{5 n+7}\right)^{n}\)b) \(\sum_{n=0}^{\infty} \frac{\pi^{n}}{n !}\)
1. Determine whether the series converges or diverges.$$ \sum_{k=1}^{\infty} \frac{\ln (k)}{k} $$convergesdiverges2.Test the series for convergence or divergence.$$ \sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{3 \pi}{n}\right) $$convergesdiverges
The radius of convergence of the power series is The radius of convergence of the power series 2. ** In is Select the correct answer. YOU MUST SHOW WORK ON SCRA 1 none of the above 2 0
Find the periodic solutions of the differential equations \((a) \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{ky}=\mathrm{f}(\mathrm{x}),(b) \frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{d}^{3} \mathrm{x}}+\mathrm{ky}=\mathrm{f}(\mathrm{x})\)where \(k\) is a constant and \(f(x)\) is a \(2 \pi\) - periodic function.Consider a Fourier series expansion for \(f(x)\) using the complex form, \(f(x)=\sum_{n=-\infty}^{n=+\infty} f_{n} e^{i n x}\) and try a solution of the form \(y(x)=\sum_{n=-\infty}^{n=+\infty} y_{n} e^{i n x}\)
Convergence of a Power Series The of a power series is the set of all values of x for which the series converges. Consider C -a)". Let R be the radius of convergence of this series. There are neo only three possibilities: 1. The series converges only when x = a, and so R = 0 and the interval of convergence is {a}. 2. The series converges for all x, and so R= oo and the interval of convergence а...
Consider the power series Find the radius of convergence R. If it is infinite, type "infinity" or "inf". Answer: R= What is the interval of convergence? Answer (in interval notation): | (1 point) Library/Rochester/setSeries8Power/eva8_6c.pg The function f(x) = is represented as a power series f(x) = cnx". Find the first few coefficients in the power series. co= || C1 = || || C4 = Find the radius of convergence R of the series. R=1
Show that the radius of convergence of any power series is given by Σ2 .Π. an 1Ξ0 R lim inf an nroo
Consider the power series k=1 (2-x+1)* 2. R . (a) Find the center of convergence of the series. (b) Find the radius of convergence of the series.