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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.) \(I=\)
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) \(\sum_{n=0}^{\infty} \frac{2^{n}(x-2)^{n}}{3 n}\)
15. . 16 . 17 Find a power series representation for the function. х f(x) (1 + 6x)2 f(x) = ( (-6).*- 1 nxt n = 0 x Determine the radius of convergence, R. R = 1/6 Evaluate the indefinite integral as a power series. t Vi dt 1 - 79 C+ Σ Σ( n = 0 What is the radius of convergence R? R= Use a power series to approximate the definite integral, I, to six decimal places. x3...
Find a power series representation for the function. f(x) = فيه (x – 4)2 00 f(x) = Σ no Determine the radius of convergence, R. R = Evaluate the indefinite integral as a power series. Je at c+ Σ ΦΟ η = Ο What is the radius of convergence R? R = Find the radius of convergence, R, of the series. 3n Σ n! n=1 R= Find the interval, 1, of convergence of the series. (Enter your answer using interval...
Show that if p and q are positive integers, then the radius of convergence of the power series below is R= +∞\(\sum_{\mathrm{k}=2}^{+\infty} \frac{(\mathrm{k}+\mathrm{p}) !}{\mathrm{k} !(\mathrm{k}+\mathrm{q}) !} \mathrm{x}^{\mathrm{k}}\)
Find the radius of convergence, R, of the series. (-1)"x Σ Find 00 n n = 1 R = Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I = [-/1.04 Points] DETAILS SCALCET8 11.8.014. Find the radius of convergence, R, of the series. 00 x8n n! n = 1 R= Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I = OFI Find the radius of convergence,...
1. Consider a geometric series of \(\sum_{n=1}^{\infty} g r^{n-1}\). Plot the \(\mathrm{n}\) -th term \(a_{n}\), and the partial sum \(s_{n}\) versus \(n\) under the following two conditions.i) with \(g=12\), and \(\mathrm{r}=2 / 3\),ii) with \(g=12\), and \(r=3 / 2\).Which of the (i) or (ii) converges? For the convergent one, approximate the total sum, i.e. \(\sum_{n=1}^{\infty} g r^{n-1}\) from your plot and draw a line on the plot to demonstrate this.2. Using the following inequality, find the value of \(\sum_{n=1}^{\infty} \frac{1}{n^{2}}\)...
3. . 4. 6. Find the radius of convergence, R, of the series. x2+4 5n! n = 1 R= Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I = Find the radius of convergence, R, of the series. xn n469 n = 1 R= Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) = Find the radius of convergence, R, of the series. Ü (x – 9)"...
11 . 12 13 Find a power series representation for the function. (Give your power series representation centered at x = 0.) f(x) = 5 x f(x) = ;- n = 0 Determine the interval of convergence. (Enter your answer using interval notation.) Find a power series representation for the function. (Give your power series representation centered at x = 0.) x2 f(x) X4 + 16 f(x) = Σ |(-1)" ) n = 0 X (a) Use differentiation to find...
In the following, we will tse a kmown power series to approximate 1/2 arctan(r) dr to within 0.00001 of the actual value of the definite integral (a) [2pt] Use a known power series representation to express (ctan(x) as a Maclaurin series. What is the radius of the series convergence? 1/2 (b) [4pts] Use your answer from part (a) to express(r) dr as an alternating series (c) [6pts] Your series in part (b) will converge by the Alternating Series Test. (You...
16) Approximate the definite integral using power series. If the antiderivative obtained is an alternating series, use the Alternating Series Estimation Theorem to ensure the error is less than 0.001; otherwise, use at least four nonzero terms to approximate the integral. (a) { er at 6) ſ'cos(x) dx