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MTH 253 - Spring Term 2006
Test 2 - No Calc Portion Name
All work on this test will be evaluated for your style of presentation as well as for the "correctness" of your "answer." Follow the writing guidelines established during lecture and spelled out in your Test 2 guide. To receive full credit all relevant algebra steps must be shown on the paper.
- Perform and absolute ratio test on the series (^ ) ( )
2
1
k k
k k
∞
∑ (^) + and state an appropriate conclusion. (10 points)
- Determine and state whether each given series is absolutely convergent, conditionally convergent, or divergent. Simply state your conclusion – no other work should be shown. (You may want to do some work on your scratch paper to help you decide your answer.) (14 points total)
` ( ) 1 2
k
k k
∞
∑ (^) + is.
1
k
k
k k
∞
∑ is^.
( )^3 1
k
k k
∞
∑ is^.
( ) ( )
1 1
ln 1
k k k
∞^ +
∑ (^) + is.
( ) 1
sin k
k k
∞ π ∑= is.
( ) 1 (^ )
k
k k k
∞
∑ (^) + is.
( ) 1
k
k k
k k
∞
∑ (^) ⋅ + is.
- a. Use the binomial series formula to find a 3 rd^ degree Taylor polynomial for the function
( ) ( ) (^2) ( )^2
f x x (^) x
. (8 points)
b. Looking at your answer to part (a), what would be the Taylor series for ( ) ( )^2
f x x
centered at x = 5? (3 points)
c. What is the interval of convergence for the series in part (b)? (4 points)
- True or False. (8 points total)
T or F If (^) k lim → ∞ ak = 0 , then the series k 1^ k
a
∞ ∑= must converge.
T or F If (^) k lim → ∞ ak = 0 , then the series (^) ( ) 1
1 k k k
a
∞
∑ (^) ⎣ ⎦must converge.
T or F If ak > 0 ∀ k ≥ 1 and k 1^ k
a
∞
∑ diverges, then the series^ (^ ) 1
1 k k k
a
∞
∑ (^) ⎣ ⎦ must also diverge.
T or F If
1 lim^1
k k k (^) k k
a x x a x
→ ∞ = ∞ ∀^ , then the interval of convergence for the power series
0 k^ k k
a x
∞
∑ is^ (^ ∞ ∞,^ ).
T or F The series (^ )^ (^ ) 1
k k
k k^ k
∞
∑ (^) ⎢⎣ + ⎥⎦ is an alternating series.
T or F The series (^ )^ (^ )^
1
1
k k
k k^ k
∞^ +
∑ (^) ⎢⎣ + ⎥⎦ is an alternating series.
T or F The series (^ )^ (^ ) 1
k k k k^ k
∞
∑ (^) ⎢⎣ + ⎥⎦ is an alternating series.
T or F The series (^ )^ (^ )^
1 1
k k k k^ k
∞^ +
∑ (^) ⎢⎣ + ⎥⎦ is an alternating series.
MTH 253 - Spring Term 2006
Test 2 - Calc Portion Name
All work on this test will be evaluated for your style of presentation as well as for the "correctness" of your "answer." Follow the writing guidelines established during lecture and spelled out in your Test 2 guide. To receive full credit all relevant algebra steps must be shown on the paper.
- Use the Taylor command on your calculator to find the 3 rd^ degree Maclaurin series of the function f (^) ( x (^) ) = sin( π x ). Write down the result. (3 points)
- Write down (through the 4 th^ digit after the decimal point) each partial sum ( S 1 (^) , S 2 (^) , S 3 ,…) of
the series (^ ) 1
1 k k kk
∞
∑ until you have established the value of^
( ) 1
1 k k kk
∞
∑ accurate through the 3^ rd digit after the decimal point. State this value and how you know that you’ve established said value. (12 points)
3. Use the Maclaurin series formula ( ) (^ )
2 1 0
sin 2 1!
k (^) k k
u u k
∞ (^) +
to help you find the Maclaurin
series for ( )
1/ 2 (^5 )
∫ 0 x^ sin x^ dx.^ Do not evaluate or estimate the series.^ (12 points)
