Question

Let f : R \ {1} → R be given by f(x) = 1 1 − x . (a) Prove by induction that f (n) (x) = n! (1 − x) n for all n ∈ N. Note: f (n) (x) denotes the n th derivative of f. You may use the usual differentiation rules without further proof. (b) Compute the Taylor series of f about x = 0. (You must provide justification by relating this specific Taylor series to general Taylor series.) (c) Determine the interval of convergence of the Taylor series for f(x). Justify your conclusions with appropriate convergence tests or facts from this subject. (d) At which point(s) x ∈ R does the Taylor series converge absolutely? At which point(s) does it converge conditionally? Justify your answer. (e) Using your result in part (a) (or otherwise) use a Taylor series to show that log 2 > 11/12.

Answer #1

You can check that log(2) is not grater than (11/12) by using a calculator! It will be (7/12).

1.
(a) Let f(x) = exp(x),x ∈ R. Show that f is invertible and
compute the derivative of f−1(y) in terms of y. [5]
(b) ﬁnd the Taylor series and radius of convergence for g(x) =
log(1+ x) about x = 0. [6]

For the series ∑∞ n=0 ((-1)^(n-1)) ((x-7)^n)/n
a) Find the radius and interval of absolute convergence.
b) For what values of x does the series converge
conditionally?

let
f(x)=ln(1+2x)
a. find the taylor series expansion of f(x) with center at
x=0
b. determine the radius of convergence of this power
series
c. discuss if it is appropriate to use power series
representation of f(x) to predict the valuesof f(x) at x= 0.1, 0.9,
1.5. justify your answe

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=

Consider the Taylor Series for f(x) = 1/ x^2 centered at x =
-1
a.) Express this Taylor Series as a Power Series using summation
notation.
b.) Determine the interval of convergence for this Taylor
Series.

1) find the Taylor series expansion of
f(x)=ln(x) center at 2 first then find its associated radius of
convergence.
2) Find the radius of convergence and interval
of convergence of the series Σ (x^n)/(2n-1) upper infinity lower
n=1

For the next two series, (1) find the interval of convergence
and (2) study convergence at the end points of the interval if any.
Also, (3) indicate for what values of x the series converges
absolutely, conditionally, or not at all. You must indicate the
test you use and show the interval of convergence both analytically
and graphically and summarize your results on the picture.
∑∞ n=1 ((−1)^n−1)/ (n^1/4)) *x^n

Consider the function f : R 2 → R defined by f(x, y) = 4 + x 3 +
y 3 − 3xy.
(a)Compute the directional derivative of f at the point (a, b) =
( 1 2 , 1 2 ), in the direction u = ( √ 1 2 , − √ 1 2 ). At the
point ( 1 2 , 1 2 ), is u the direction of steepest ascent,
steepest descent, or neither? Justify your...

find the taylor series f(x)=1/x at c=1 and the radius of
convergence.

Apply term-wise integration to the expansion 1/(1 − x) = ∑∞ n=0
x n = 1 + x + x^2 + x^3 + ... to prove that for −1 < x < 1, −
ln(1 − x) = ∑∞ n=0 (x^n+1)/(n + 1) = x + x^2/2 − x^3/3 + x^4/4 +
... You should find a constant that appears when you integrate. (b)
Study convergence of this new series at the end points of the
interval (−1, 1). (c)...

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