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)=x−12x2+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).
- ln(1−x2)
- x1+x2
- arctan(x3)
- 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(x−a)n). n=0 an (x−a)n). [Hint: sinx=sin(a+x−a)].
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(x−a)n.
- lnx,a=1
- ex,a=3
- x3+2x2+3,a=1
- 1x,a=5
Q6
Evaluate the following integrals as series.
- ∫1x=0ex2dx
- ∫1x=011+x4dx
- ∫1x=03√1−x3dx