Find the sum of the infinite series whose sequence of partial sums, Sn, is S sub n = 3 - 1/n + 100 (5 points) I got 0, is this right? or is it 3.
A) 0
B) 3
C) 2.99
D) Sum does not exist
as written Sn diverges, since the nth term is 103 - 1/n
However, assuming the usual carelessness with parentheses,
Sn = 3 - 1/(n+100) → 3
How did you get 0? Surely not by using (3-1)/(n+100)
To find the sum of an infinite series, we need to examine the behavior of the sequence of partial sums (S sub n) as n approaches infinity. Let's start by calculating some terms of the sequence for different values of n:
For n = 1, S sub 1 = 3 - 1/1 + 100 = 102
For n = 2, S sub 2 = 3 - 1/2 + 100 = 102.5
For n = 3, S sub 3 = 3 - 1/3 + 100 = 102.67
For n = 4, S sub 4 = 3 - 1/4 + 100 = 102.75
...
As we can see, the values of the sequence of partial sums are increasing and approaching a limit. Therefore, the sum of the series can be determined by finding the limit of the sequence as n approaches infinity.
lim(n→∞) S sub n = lim(n→∞) (3 - 1/n + 100)
To evaluate this limit, we can focus on the dominant term, which is 1/n. As n approaches infinity, the value of 1/n becomes infinitely small and tends towards zero. As a result, we can effectively ignore this term:
lim(n→∞) (3 - 1/n + 100) ≈ lim(n→∞) (3 + 100) = 103
Therefore, the sum of the infinite series is approximately 103. Since none of the given answer choices matches this value, we can conclude that the sum does not exist. So the correct answer is:
D) Sum does not exist