nn is convergent. Deduce that lim (2n)! nn 6. Show that the series = ) (2n)!...
Show that the series \(\sum_{n=1}^{\infty} \frac{n^{n}}{(2 n) !}\) is convergent. Deduce that \(\lim _{n \rightarrow \infty} \frac{n^{n}}{(2 n) !}=0\)
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nn is convergent. Deduce that lim (2n)! nn 6. Show that the series = ) (2n)!...
If the series
If the series \(\sum_{n=1}^{\infty} a_{n}\) converges and \(a_{n}>0\) for all \(n\), which of the following must be true?(A) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0\)(B) \(\left|a_{n}\right|<1\)for all \(n\)(C) \(\sum_{n=1}^{\infty} a_{n}=0\)(D) \(\sum_{n=1}^{\infty} n a_{n}\) diverges.(E) \(\sum_{n=1}^{\infty} \frac{a_{n}}{n}\) converges.
Determine whether the series is convergent or divergent.
Determine whether the series is convergent or divergent.$$ \sum_{n=1}^{\infty}\left(\frac{8}{e^{n}}+\frac{4}{n(n+1)}\right) $$convergentdivergentIf it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
find an expression for the area of the region under the graphf(x)=x^4 on the interval...
find an expression for the area of the region under the graph f(x)=x^4 on the interval [1,7]. use right-Hand endpoints as sample points choices1. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{7 i}{n}\right)^{4} \frac{7}{n}\)2. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{9 i}{n}\right)^{4} \frac{6}{n}\)3. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{6 i}{n}\right)^{4} \frac{6}{n}\)4. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{7 i}{n}\right)^{4} \frac{6}{n}\)5. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{6 i}{n}\right)^{4} \frac{7}{n}\)6. area \(=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{9 i}{n}\right)^{4} \frac{7}{n}\)
7. Use the Alternating Series Test to determine the convergence or divergence of the series
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) \(\sum_{n=0}^{\infty} \frac{\pi^{n}}{n !}\)
Determine whether the series converges or diverges.
Determine whether the series converges or diverges.(1) \(\sum_{n=1}^{\infty} \frac{e^{1 / n}}{n^{2}}\)(2) \(\sum_{n=1}^{\infty}\left(\frac{2}{\sqrt{n}}+\frac{(-1)^{n}}{3^{n+1}}\right)\)(3) \(\sum_{n=1}^{\infty} \frac{5-2 \sin n}{n}\)(4) \(\sum_{n=1}^{\infty} \frac{3+\cos n}{n^{3 / 2}}\)(5) \(\sum_{n=0}^{\infty} \frac{\sqrt{n^{2}+2}}{n^{4}+n^{2}+5}\)(6) \(\sum_{n=1}^{\infty=1}\left(1+\frac{1}{n}\right)^{n}\)(7) \(\sum_{n=1}^{\infty} \frac{n+1}{n 2^{n}}\)(8) \(\sum_{n=1}^{\infty} \frac{\arctan n}{n^{4}}\)(9) \(\sum_{n=1}^{\infty} n \sin \frac{1}{n}\)
Determine whether the series converges, and if so, find its sum.
Determine whether the series converges, and if so, find its sum. (1) \(\sum_{n=1}^{\infty} 3^{-n} 8^{n+1}\)\((2) \sum_{n=2}^{\infty} \frac{1}{n(n-1)}\)(3) \(\sum_{n=0}^{\infty}(-3)\left(\frac{2}{3}\right)^{2 n}\)(4) \(\sum_{n=1}^{\infty} \frac{1}{e^{2 n}}\)(5) \(\sum_{n=1}^{\infty} \ln \frac{n}{n+1}\)(6) \(\sum_{n=1}^{\infty}[\arctan (n+1)-\arctan n]\)(7) \(\sum_{n=1}^{\infty} \ln \left(\frac{n^{2}+4}{2 n^{2}+1}\right)\)(8) \(\sum_{n=1}^{\infty} \frac{1+2^{n}}{3^{n}}\)(9) \(\sum_{n=1}^{\infty}\left[\cos \frac{1}{n^{2}}-\cos \frac{1}{(n+1)^{2}}\right]\)
(2 points) The area A of the region S that lies under the graph of the...
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
Determine whether the given series converges or diverges
Determine whether the given series converges or diverges. Fully justify your answe(a) \(\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} \ln n}\)(b) \(\sum_{n=1}^{\infty} \cos \left(\frac{1}{n^{2}}\right)\)(c) \(\sum_{n=1}^{x} \frac{(2 n) !}{5^{n} n ! n t}\)
Determine whether the series converges or diverges.
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
Cansider the series
Cansider the series \(\sum_{n=1}^{\infty} a_{n}\) where$$ a_{n}=\frac{\left(6 n^{2}+2\right)(-7)^{n}}{5^{n+1}} $$In this problem you must attempt to use the Ratio Test to decide whether the series converges.Compute$$ L=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| $$Enter the numerical value of the limit \(L\) if it converges, INF if it diverges to infinity, -INF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity.L= _______Which of the following statements is true?A. The Ratio Test says that the series converges absolutely.B. The...
If the series \(\sum_{n=1}^{\infty} a_{n}\) converges and \(a_{n}>0\) for all \(n\), which of the following must be true?(A) \(\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|=0\)(B) \(\left|a_{n}\right|<1\)for all \(n\)(C) \(\sum_{n=1}^{\infty} a_{n}=0\)(D) \(\sum_{n=1}^{\infty} n a_{n}\) diverges.(E) \(\sum_{n=1}^{\infty} \frac{a_{n}}{n}\) converges.
(2 points) The area \(A\) of the region \(S\) that lies under the graph of the continuous function \(f\) on the interval \([a, b]\) is the limit of the sum of the areas of approximating rectangles:$$ A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x $$where \(\Delta x=\frac{b-a}{n}\) and \(x_{i}=a+i \Delta x\).The expression$$ A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right) $$gives the area of the function \(f(x)=\) on...
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