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Math 206 - Calculus II
Homework due October 28
In class, we learned that to truly understand the infinite series
∑^ ∞
k=
ak = a 0 + a 1 + a + 2 + · · · + an + · · ·
we must first understand the n-th partial sum
∑^ n
k=
ak = a 0 + a 1 + · · · + an
and then take a limit of these partial sums as n → ∞.
Question 1. Consider the sequence
∑^ ∞
k=
k!
(a) Find the first 6 partial sums. To do this, you will need to use the interesting fact that 0! = 1. When adding, please keep at least six decimal places. Thus, you need to compute the following seven sums:
∑^0
k=
k!
∑^1
k=
k!
∑^2
k=
k!
∑^3
k=
k!
∑^4
k=
k!
∑^5
k=
k!
∑^6
k=
k!
(b) As your partial sum
∑^ n
k=
k! increases, compare its value to that of the
natural number e ≈ 2. 718281828 ...
(c) Do you think that
∑^ ∞
k=
k!
converges or diverges? If it converges, to what
does it converge?
In class, we saw an example of a telescoping sum, where when we considered the n-th partial sum, many interior terms cancelled out. Then, we were able to compute the entire infinite series by taking a limit of the n-th partial sum as n → ∞.
Question 2. Consider the following telescoping series. Use the given hint to re-write the n-th partial sum ∑n k=1 ak^ in a simpler form by canceling most terms, then take a limit as n → ∞ to find the infinite series
k=1 ak.
(a)
∑^ ∞
k=
k^2 − k
Hint:
k^2 − k
k − 1
k
(b)
∑^ ∞
k=
k^2 + 2k Hint:
k^2 + 2k
k
k + 2
. You may have to write
out several of the first and last terms to see what cancels and what doesn’t cancel.
Recall that in class we learned about the Divergence Test, which stated the following.
Divergence Test. If ak 6 → 0, then
∑^ ∞
k=
ak diverges.
Thus, if we know that ak converges to some non-zero limit or that it diverges, then the series
k=0 ak^ will diverge as well.
Question 3. Consider the following series. When possible, use the Diver- gence Theorem to conclude that the given series diverges. If you cannot use the Divergence Theorem to conclude divergence, note that as well.
(a)
∑^ ∞
k=
k^2 − k 7 k^2 + 1
(b)
∑^ ∞
k=
(−1)kk
(c)
∑^ ∞
k=
k + 1
(d)
∑^ ∞
k=
3 k^4 + 7k^2 5 k^6
