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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?

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Missouri State University

Campbell University

Oregon State University

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

08:34

Let $f_{n}(x)=(\sin n x) /…

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07:16

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03:49

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01:28

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02:04

Find the values of $x$ for…

01:34

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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