Question

(a) Find the Maclaurin series expansion for f(x) = e 3x and prove that it converges for all x ∈ R.

(b) Decide whether the series X∞ n=1 n + 1/ sqrt (n + 2)(n + 3)(n + 4) converges or diverges( (n + 2)(n + 3)(n + 4) is all under the sqrt)

Answer #1

Find the Maclaurin series expansion for the function.
f(x)=e^6x

find the maclaurin series of f(x)=5sin(3x)

find the maclaurin series for f(x)= (eX)-(e-x)/(2)

f(x) = e x ln (1+x) Using the table of common Maclaurin
Series to find the first 4 nonzero term of the Maclaurin Series for
the function.

Find the first five nonzero terms of the Maclaurin
expansion.
f(x) = e^x/(1+x)

For the following functions, find the Maclaurin series:
f(x)=e^xsinx.

How can I prove that the Maclaurin series of (1+x)^k equals to
the binomial expansion with using Taylor’s Inequality? (NOT using
Ratio Test.)

1. The Taylor series for f(x)=x^3 at 1 is ∞∑n=0 cn(x−1)^n.
Find the first few coefficients.
c0=
c1=
c2=
c3=
c4=
2. Given the series:
∞∑k=0 (−1/6)^k
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
∞∑k=0 (−1/6)^k=

Determine if each of the following series converges or diverges
showing all the work including all the tests used. Find the sum if
the series converges.
a. Σ (n=1 to infinity) (3^n+1/ 7^n)
b. Σ (n=0 to infinity) e^n/e^n + n

Find a Fourier Series expansion for the function f(x)= xcos(3x)
on the domain from x = [-pi,pi]

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