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Mathematics LibreTexts

2.E: Calculus in the 17th and 18th Centuries (Exercises)

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Q1

Use the geometric series to obtain the series

ln(1+x)=x12x2+13x3=n=0(1)nn+1xn+1

Q2

Without using Taylor’s Theorem, represent the following functions as power series expanded about 0 (i.e., in the form n=0anxn).

  1. ln(1x2)
  2. x1+x2
  3. arctan(x3)
  4. ln(2+x) [Hint: 2+x=2(1+x2)]

Q3

Let a be a positive real number. Find a power series for ax expanded about 0. [Hint: ax=eln(ax)].

Q4

Represent the function sinx as a power series expanded about a (i.e., in the form n=0an(xa)n). n=0 an (x−a)n). [Hint: sinx=sin(a+xa)].

Q5

Without using Taylor’s Theorem, represent the following functions as a power series expanded about a for the given value of a (i.e., in the form n=0an(xa)n.

  1. lnx,a=1
  2. ex,a=3
  3. x3+2x2+3,a=1
  4. 1x,a=5

Q6

Evaluate the following integrals as series.

  1. 1x=0ex2dx
  2. 1x=011+x4dx
  3. 1x=031x3dx

This page titled 2.E: Calculus in the 17th and 18th Centuries (Exercises) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Eugene Boman and Robert Rogers (OpenSUNY) via source content that was edited to the style and standards of the LibreTexts platform.

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