University:
San Diego State University
Course:
MATH 151 | Calculus II
Academic year:

2023
Views:

200

Pages:

1
Author:

Bailey Hester

Workshop #21 1. Consider the series 1 + 1 1 1 + + + · · ·. 3 9 27 (a) The fifth partial sum is s5 = 1 + the expression 13 s5 . 1 1 1 1 + + + . Write out the terms that would appear in 3 9 27 81 (b) Looking at what you have written above, what terms would be left in the expression s5 − 13 s5 ? (c) Use the above expressions to determine what the numeric value of s5 is. (Do not just add up the first five terms.) 1 1 1 1 (d) The nth partial sum is sn = 1 + + + · · · + n−2 + n−1 . Write out the first few and last 3 9 3 3 few terms that would appear in 13 · sn . (e) Looking at what you have written above, what terms would be left in the expression sn − 13 sn ? (f) Use the above expressions to determine a formula for sn . (g) What is the limit as n goes to infinity of sn ? 2. Consider the series 2 + 2 2 2 + + + · · ·. 5 25 125 (a) The fifth partial sum is s6 = 2 + appear in the expression 51 s6 . 2 2 2 2 2 + + + + . Write out the terms that would 5 25 125 625 3125 (b) Looking at what you have written above, what terms would be left in the expression s6 − 15 s6 ? (c) Use the above expressions to determine what the numeric value of s6 is. (Do not just add up the first six terms.) 2 2 2 2 2 (d) The nth partial sum is sn = 2 + + + + · · · + n−2 + n . Write out the first few and 5 25 125 5 5 last few terms that would appear in 51 · sn . (e) Looking at what you have written above, what terms would be left in the expression sn − 15 sn ? (f) Use the above expressions to determine a formula for sn . (g) What is the limit as n goes to infinity of sn ? 3. (a) In problem #1, why did we multiply sn by 13 ? (b) In problem #2, why did we multiply sn by 15 ? (c) For which of the following series could we use a method similar to the above two problems in order to find the sum of the series? 1 1 1 + + + ··· 2 3 4 5 5 5 5 + + + + ··· 3 6 12 24 1 2 3 4 + + + + ··· 2 3 4 5 1 2 4 8 + + + + ··· 3 9 27 81 1+ (d) Find the sum of the series you indicated above.