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A review for the final exam of math 106, focusing on topics such as second-degree taylor polynomials, series convergence tests, and integration. Students are guided through estimating values using taylor polynomials, determining convergence or divergence of series using various tests, and evaluating integrals. Key concepts include the nth term test, geometric series, alternating series, comparison test, integral test, and p-series.
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Math 106: Review for Final Exam, Part II
(a)
1
7 + 5 sin x x^2
dx
(b)
1
1 + 3x^2 + 2x^3 √ (^310) x (^12) + 17x 10 dx
(a) ak = 3 +
10 k
(b) ak = (−1)k
(c) ak =
3 + 5k 7 + 2k
Strategy. The following is a good order in which to consider the various series convergence tests.
(a) Do the individual terms approach 0? If they don’t approach 0, the nth Term Test tells you the series must diverge. If they do approach 0, try another test. (b) Is the series geometric? (That is, do you multiply by the same constant r to get from each term to the next?) If so, the series converges if |r| < 1 and diverges otherwise. (c) Does the series contain something such as (−1)k^ or (−1)k+1^ or cos (kπ) that makes its terms alternate? If so, try the Alternating Series Test. (d) Does the series contain a factorial (k!) or exponential (such as 2k^ or ek)? If so, try the Ratio Test. (e) If the series has positive terms, does it remind you of a simpler series (especially a p-series: powers of k such as 1/k or 1/k^2 )? If so, try the Comparison Test. (f) Is the formula something you can integrate easily? If so, try the Integral Test.
(a) 3.1 + 3.01 + 3.001 + 3.0001 + ...
(b) 1 + 1/2 + 1/3 + 1/4 + ...
(c) 5 − 5 /3 + 5/ 9 − 5 /27 + ...
k=
(x + 3)k k · 5 k^
(a) 1 − 1 +
(b)
(c) 1 −
π^2 2
π^4 24
π^6 720
0
e−x
2 dx and show that it converges.
(b) If f(x) = e−x
2 , what is f(400)(0)? What is f(401)(0)?
