Calculus 1301B Winter 2023 Written Assignment 2
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Calculus 1301B Winter 2023
Written Assignment 2
1. Show that, if P(x) and Q(x) are polynomials of positive degrees, then the sequence an = rnI I converges to 1.
[Hint: First show that the n’th root of the absolute value of any non-zero polynomial in n converges to 1, by applying the Squeeze Theorem and the fact that limn!1 n = 1 = limn!1 c, for any constant c > 0.]
2. Let (an ) be a sequence with positive terms, such that nlim!1 = 2023. Show that nlim!1an = 1. [Hints:
– Recall that lim cn = L, when for every e > 0 there exists an integer N such that L−e < cn < L+e
for all n ≥ N . Use this definition to show that there exists an integer N such that an+1 ≥ an for all n ≥ N .
– Suppose that (an ) is bounded above. Can you then conclude something about its limit?
– Apply the Algebraic Limit Theorem to .]
3. Find the limits of the following sequences. Justify your answers.
r
{z
n times
ln(n + 1) − ln n
lnn · ln(n + 1) .
4. Determine whether the following series are absolutely convergent, conditionally convergent, or diver- gent. Justify your answers.
(a) (−1)n
(b) (−1)n
(c) (−1)n
(d) (−1)n .
5. Determine for what values of the variable x is the following series convergent ✓ ◆n . Justify your answer.
6. Find the interval of convergence of the power series (1 + + ··· + ) · xn . Justify your answer. [Hint: Suppose first that the variable x satisfies |x| < 1, and consider the ratio |an+1|/|an |, where an = (1 + 1/2 + ··· + 1/n) · xn . Try to find an upper bound for |an+1|/|an | of the form bn · |x|, with
|an+1 |
n!1 n!1 |an |
does not converge to zero if |x| ≥ 1.]
2023-03-08