Series Approximations

The questions below are due on Thursday February 13, 2025; 02:00:00 PM.

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This lab explores series representations of mathematical functions using numerical and analytic methods. You should use python3 for computations and matplotlib to make plots, as illustrated by the following example:

from math import sin, pi
from matplotlib.pyplot import plot, show
x = []
y = []
t = 0
while t<1:
    x.append(t)
    y.append(sin(2*pi*t))
    t += 0.01
plot(x,y)
show()

Part a: Infinite series. Determine a closed-form expression (i.e., no infinite sums) for the following function:

f(t)=\sum_{k=1}^\infty{\sin(kt)\over k}

Express your answer as a mathematical expression or by making a plot.

Part b: Even-numbered terms. Determine a closed-form expression for the series that includes only the even-numbered terms of the series in the previous part:

f_e(t)=\sum_{\mbox{k=2}\atop\mbox{k even}}^\infty{\sin(kt)\over k}

Express your answer as a mathematical expression or by making a plot.

Describe the relation between this answer and the answer to the previous part.

Briefly explain why this relation holds.

Part c: Odd-numbered terms. Determine a closed-form expression for the series that includes only the odd-numbered terms of the series in the previous part:

f_o(t)=\sum_{\mbox{k=1}\atop\mbox{k odd}}^\infty{\sin(kt)\over k}

Express your answer as a mathematical expression or by making a plot.

Describe the relation between this answer and the answers to the previous parts.

Briefly explain why these relations hold.

Check In:
Please describe your approach to each of the parts (a, b, and c) of this lab.

Please upload your solutions to all parts of this problem (a, b, and c) as a single pdf file:

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