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MATH 152
Python Lab
Math 152 – Python Lab 6
Directions: Use Python to solve each problem. (Template link)
1. Given the sequence an =
1 +
1
n2
:
(a) Evaluate a1, a2, ..., a10 to numerically estimate the limit.
(b) Plot the first 30 points of the sequence to graphically estimate.
(c) Use Python to calculate limn→∞
an.
2. Given bn =
(4/5)n
n
:
(a) Show that limn→∞
bn = 0.
(b) Plot the first 50 terms of the sequence and the first 50 partial sums on the same graph.
(c) Evaluate the exact and approximate sum of the series.
3. Given the series X
n=1
an, where an =
3
n2 + 4n
:
(a) Evaluate limn→∞
an
(b) Write the first 12 terms of the sequence and the first 12 partial sums of the series.
(c) Calculate the partial fraction decomposition of an.
(d) Based on your answer in (c), state a general formula for sn, the nth partial sum, and find
the sum of the series or show it diverges.
4. Given a0 = 1 and an+1 =
1
3a
2
n
+
2an
3
(a) Evaluate the first ten terms of the sequence. Evaluate the limit as n → ∞.
(b) Repeat the process given a0 = 100.
(c) Based on parts (a) and (b), can you say anything about the sequence behavior regardless
of the starting term?
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