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MATH 106 Final Exam Review, Part II
- The heights of women in a certain country are normally distributed with a mean of 170 cm and a standard deviation of 10 cm. What fraction of these women have heights between 165 cm and 180 cm?
- Use a second-degree Taylor polynomial to estimate 3
- What is the largest possible error that could have occurred in your previous estimate?
- Use a comparison to show whether each of the following converges or diverges. If an integral converges, give a good upper bound for its value.
(a)
1
7 + 5 sin x x^2
dx
(b)
1
1 + 3x^2 + 2x^3 โ (^310) x (^12) + 17x 10 dx
- Decide if each of the following sequences {ak}โ k=1 converges or diverges. If a sequence converges, compute its limit.
(a) ak = 3 +
10 k
(b) ak = (โ1)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 not, the nth Term Test tells you the series must diverge. (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 (โ1)k^ or (โ1)k+1^ or sin (kฯ/2) 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.
- Decide if each of the following series converges or diverges. If a series converges, find its value.
(a) 3.1 + 3.01 + 3.001 + 3.0001 + ...
(b) 1 + 1/2 + 1/3 + 1/4 + ...
(c) 5 โ 5 /3 + 5/ 9 โ 5 /27 + ...
- Decide if each of the following series converges or diverges. If a series converges, find upper and lower bounds for its value.
(a)
โ^ โ
k=
(โ1)k โ (^3) k + 1
(b)
โ^ โ
k=
(2k)! 3 k^ (k!)^2
- Find the complete Taylor series (in summation notation) for f(x) = ln (1 โ x) about x = 0 and determine its interval of convergence.
- Write the complete series equal to
0
eโx
2 dx and show that it converges.
