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Course: Calculus, all content (2017 edition)>Unit 7

Lesson 12: Power series intro

Integrating power series

Within its interval of convergence, the integral of a power series is the sum of integrals of individual terms: ∫Σf(x)dx=Σ∫f(x)dx. See how this is used to find the integral of a power series.

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• Apart from its just a problem. Can anyone tell, what is the integration of power series means or applications of differentiating/integrating a power series?
• It's also useful for Fourier series, and the Fourier series are useful to transform a function of time to a function of frequency and functions of frequency are useful in signal analysis and signal analysis....
• Why didn't we prove that the bounds 0 to 1 were within the interval of convergence?
• I guess you just have to assume, although if you do it out, it is in fact within the interval of convergence (-4,4).
• I just got confused at the end where he divided the first term 1/16 by 1 - 1/4.
How did he arrive to that conclusion?
• 𝐺 = 𝑎 + 𝑎𝑟 + 𝑎𝑟^2 + 𝑎𝑟^3 + ... + 𝑎𝑟^(𝑛 − 1)
is a geometric sum with 𝑛 terms.

We realize that
𝐺 − 𝑟𝐺 = 𝑎 − 𝑎𝑟^𝑛 ⇔ 𝐺 = 𝑎(1 − 𝑟^𝑛)∕(1 − 𝑟)

With |𝑟| < 1 we get
lim(𝑛→∞) 𝐺 = 𝑎∕(1 − 𝑟)
• Why are integrate term by term in power series?
• Say we have an indefinite integral of a sum (a + b). In this case we can evaluate this integral as a sum of two integrals. In other words; integral of a+b equals itegral of a + integral of b.
Same reasoning can be used when thinking about integrating series: integrating whole series is the same as taking series of integrals.
• When can we exchange the series symbol with the integral symbol?
• We don't exchange the symbols, rather we are integrating the series. you can never just "exchange" symbols.
• Could you explain the very end of this? Why is the 1st term divided by (1-r) to find the actual value of the integral? And just to clarify this means that 1/12 is equal to the 3rd integral of f(x) from 0 to 1 right?
• The formula for the value of a geometric series is a/(1-r), where a is the first term and r is the common ratio. Check out Khan Academy's videos of this. And this indeed means 1/12 is equal to the integral of f(x) from 0 to 1.
(1 vote)
• Can we integrate other types of series?
• Yes, as long as it can be written in Sigma notation.
(1 vote)
• can this be used to approximate the integrals of functions that are impossible/really difficult to integrate?(by writing the function as a taylor series and then integrating the series)