Answer to Question #203265 in Real Analysis for Raj kumar

Question #203265

Test the series:


n=1 ∑∞ (-1)n-1 [Sin(nx)]/n√n


for absolute and conditional convergence.


1
Expert's answer
2021-06-16T06:25:55-0400
n=1(1)nsin[nx]nn\displaystyle\sum_{n=1}^{\infin}\dfrac{(-1)^n\sin[nx]}{n\sqrt{n}}

sin[nx]1,xR,n1|\sin[nx]|\leq1, x\in\R, n\geq1

Then


(1)nsin[nx]nn1nn,xR,n1\bigg|\dfrac{(-1)^n\sin[nx]}{n\sqrt{n}}\bigg|\leq\dfrac{1}{n\sqrt{n}}, x\in\R, n\geq1

The series n=11nn\displaystyle\sum_{n=1}^{\infin}\dfrac{1}{n\sqrt{n}} converges as pp -series with p=32>1.p=\dfrac{3}{2}>1.


Therefore the series n=1(1)nsin[nx]nn\displaystyle\sum_{n=1}^{\infin}\dfrac{(-1)^n\sin[nx]}{n\sqrt{n}} converges absolutely by the Comparison Test, for xR.x\in\R.



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Comments

Assignment Expert
15.07.21, 22:10

Dear Rajkumar, please use the panel for submitting a new question.


Rajkumar
09.06.21, 16:54

Test the following series for convergence ∑ n=1 ∞ [✓n^4+9 -✓n^4-9]

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