Can someone help me with the problem set question? I need the answer for all the questions. Thanks.

MAT 137Y: Calculus!
Problem Set 10
Due on Friday, April 8 at 1pm at SS 1071
Instructions:
? Print this cover page, ?ll it out entirely, and STAPLE it to the front of
your problem set solutions. (You do not need to print the questions.)
Doing this correctly, and submitting it to the correct box (see below) is
worth 1 mark.
? Submit this problem set in the boxes labelled MAT137 at SS
1071. Submit it in your TA?s box only. It is due on Friday,
April 8, by 1pm. You may submit it earlier on that day or earlier
during the week if you prefer.
Please, double-check your tutorial code on blackboard, and double-check
your TA name on the course website. Remember that if there is a discrepancy between Blackboard and ROSI/ACORN, then your correct tutorial
is the one on Blackboard, not on ROSI/ACORN.
See http://uoft.me/137tutorials
? Before you attempt this problem set do all the practice problems from
sections 12.6?12.9.

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1. Give an example of a power series…
(a) whose interval of convergence is (?42, 42).
(b) whose interval of convergence is [?5, ?3].
(c) whose interval of convergence is [e, ?).
(d) whose interval of convergence is [?1, 1] and which is conditionally convergent
both at ?1 and at 1.
?
(e) centered at x = ? 2 and whose interval of convergence is (??, ?).
2. Find values of the constants a and n so that the limit
axn + esin x ? [1 + ln (1 + x2 )] cos x
x?0
x4

L = lim

exists and is not 0. For those values, calculate L.
3. (a) Find a function F that satis?es the following properties:
?
?
?
?

The domain of F is all the real numbers.
F (x) = cos(x2 )
F (0) = 3
F (0) = 4

You won?t ?nd an easy antiderivative for cos(x2 ). Instead, write your ?nal
answer for F as a power series.
(b) Estimate the value of F (1) with an error smaller than 0.00001.
Hint: Alternating series.
4. This question is an example of an application of Taylor series to physics, but you do
not need to know any physics to solve it.
Charged particles create a property in space called electric potential. If we have a
particle with charge q at a point P , it will create an electric potential V at a point
Q given by the equation
kq
V =
r
where k is a constant and r is the distance between P and Q. We say that the
electric potential created by one particle is inversely proportional to the distance.
(a) Let?s say that we have a charge with value q at the point x = a in the x-axis and
a charge with value ?q at the point with x = ?a. (We call this a dipole.) We
want to study how the total electric potential (that is, the sum of the electric

potentials created by both particles) depends on the distance at points very far
away from both charges. The total electric potential at a point x > a in the
x-axis will be
kq
kq
?
.
(1)
V =
x?a x+a
Since we are looking at points very far away from both charges, we may assume
a
that x is much bigger than a. Let us call u = . Then the quantity u is very
x
small.
Express Equation 1 in terms of k, q, x, and u (but not a). Then write it as a
Taylor series using u as the variable.
Hint: All you need is the geometric series, which you already know. You do
not need to take any derivatives.
(b) Since u is very small, it makes sense to keep only the ?rst non-zero term of
the Taylor series you obtained in Question 4a. Do so. If you do this correctly,
you will have proven that the potential created by these two charges together
is directly proportional to a and inversely proportional to the square of the
distance.
(c) This time assume that we have a charge with value q at x = a, a charge with
value ?2q at x = 0 and a charge with value q at x = ?a. Then the total electric
potential created by these charges is
V =?

2kq
kq
kq
+
?
.
x?a
x
x+a

Do a calculation similar to the above to answer the following question: For
values of x far away from these charges, the total electric potential is inversely
proportional to which power of the distance?