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MTH 253 - Fall Term 2006 Test 2 Name
- Each of the series on this page either converges "by inspection" or diverges "by inspection." While a convergence test would have to be done toformally establish whether the series is convergent or divergent, a person with a good understanding of this material knows what the results of any such test would be.
For each series, write the word convergent in the provided blank if a convergence test would (correctly) show the series to be convergent and write the work divergent in the provided blank if a convergence test would (correctly) show the series to be divergent. (2 points each) No explanation and/or work need be, nor should be, shown for this problem. ☺
n
n
n n
∞
1
sin 2 n
n
n
∞
( ) 1 ( ) 1
k
k
k k
∞^ +
2 2 2
k^1
k k k k
∞
∑
( ) 1 2 3 3
k k
k k
∞ (^) +
( ) 1
1 n! n n
n n
∞
( ) 1
k k k
∞
MTH 253 - Test 2
- Perform the Absolute Ratio Test for Convergence on the series ( ) (^ ) ( )
2 1
k^ k k
k k
∞
∑ ⎢⎣ + ⎥⎦ and
state the appropriate conclusion. (12 points) You must show the algebraic steps used to simplify and evaluate the limit. You must state a formal conclusion.
MTH 253 - Test 2
- Use the Sequence of Partial Sums to determine the value of (^2 ) 1 0
k k k k
∞ = +
∑ to three digits after
the decimal. State each partial sum value through three digits until you have shown enough partial sums to establish the stated accuracy. Write a formal conclusion sentence; include a brief statement as to why you knew you had the desired accuracy. (12 points)
MTH 253 - Test 2
5. Find the Taylor Series Representation for ∫ 0 1/ 2 x cosh 2( x^3 ) dx. Completely simplify the
series formula. Do not approximate the value of the integral and/or series. (12 points) Show all relevant work!
Given: ( )
2 0
cosh 2!
k k
u u u k
∞
= ∑ ∀. (The function is read “hyperbolic cosine.”)
MTH 253 - Test 2
- Find the interval of convergence for the power series (^ )^ ( ) 1
k k k k x
∞
. (20 points)
Organize your work in a meaningful manner; write complete sentences where appropriate; state your conclusion in a meaningful and emphatic manner.
MTH 253 - Test 2
8. Consider the fact that sin −^1 ⎛⎜^12 ⎞ ⎟=^ π 6
. This implies that π = 6sin −^1 ( 0.5) so we can use a Taylor Series for sin −^1 ( x )to generate digits of π.
Imagine that you were working on such a problem a couple of hundred years ago. Let’s see what
approximation for π you would come up with if you generated and used the fifth degree Taylor
polynomial, T 5 (^) ( x (^) ), for sin −^1 ( x )centered about x = 0.
a. The first 5 derivatives of f (^) ( x (^) ) = sin−^1 ( x ) are given below. Use Taylor’s formula to find T 5 ( x ). (10 points)
.
b. What is the estimate for π that you get when using T 5 (^) ( 0.5)to generate your estimate? Round your answer to the nearest 1000th^. (3 points)
