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# Let $f_n (x) = \left( \sin nx \right)/n^2.$ Show that the series $\sum f_n(x)$ converges for all values of $x$ but the series of derivatives $\sum f'_n (x)$ diverges when $x = 2n \pi, n$ an integer. For what values of $x$ does the series $\sum f''_n (x)$ converge?

## This will comverge only when $\sin (n x)=0,$ otherwise $\sin (n x)$ will oscillatebetween $-1$ and 1 as $n \rightarrow \infty .$For it to be $0, n x$ has to be an integer multiple of $\pi,$ so$x=n \pi, \quad$ where $n$ is an integer

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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the problem is like F N X is equal to sign. An X over and square showed that the series some of the NX converges for our values of X. At the series of derivatives emerges, one X is equal to two and pie and interview for what values of acts as a series second derivative converging. The first notice that the absolute value of NX is equal to the value of Sign an axe over in square, which is less than one over and score, and we have the sum of one over and square converters. So, by the comparison, Syria as a comparison test to have some of them ax covered Terrace Sylvia half some of annex calmer GIs. Mhm. Then we compute f N x brown. This is a cultural and tabs cause I and acts of our and square, which is equal to assign and acts over in one X is ego to chew and pie. We have F F and prom X, the two m pie, which is equal to co sign shoe and square. How's pie over end, which is equal to one over and on Behalf series after and shoo em pi from This is a cultural some one over in which diverges, then my computer, The second derivative, US and ax. This is a culture negative sign an axe hams and over and which is a co two negative sign. Mm x. So if the sum of n x from prom convert this So we have sign N X Must be goes to zero must goes to zero What? And goes to infinity So we have axe should have been echo two zero more Hi too high So this is Cape High Your k is an interior in this case Mhm Sign an axe sign times k pi It's always zero The series. It's just a series of zero which is equal to zero.

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

Lectures

Join Bootcamp