Using Series Approximations

The questions below are due on Monday March 01, 2021; 10:00:00 PM.

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Parts 1a, 1b, and 1c of this Lab should be completed prior to this Friday's check-in.

1) Infinite series

1a. Use Python to generate an approximation to the following function, using a finite numbers of terms in the sum. Generate a plot of your results, and use that to guess at a closed-form expression (i.e., no infinite sums) for f(t), defined as:

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

Sketch or plot this function, and upload it below.

The file type should be png, jpg, svg, pdf, or something similar.
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1b. Repeat part 1a for the function g(t) that contains just the even-numbered terms in the series for f(t):

g(t) = \sum_{{k=1}\atop k~{\rm even}}^\infty {\sin(k t)\over k}
What similarities and differences are there between the plots of f(t) and g(t)? For your check-in, be prepared to explain the differences between these functions arise, as well as how those differences arise.

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1c. Repeat part 1a for the function h(t) that contains just the odd-numbered terms in the series for f(t):

h(t) = \sum_{{k=1}\atop k~{\rm odd}}^\infty {\sin(k t)\over k}
How do the sketches of f(t) and h(t) differ? Be prepared to explain how these differences arise.

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2) Finite series

2a. Let f_N(t) represent the sum of a finite number N of terms in the series in part 1a:

f_N(t) = \sum_{k=1}^N {\sin(k t)\over k}

Upload a plot of f_N(t) when N=100.
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Describe similarities and differences between f(t) and f_N(t), and describe how the differences between f(t) and f_N(t) change as N increases. Do the differences approach zero as N approaches \infty?
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2b. Let F_N(t) represent the average of the first N finite sums from part 2a.

F_N(t) = {f_1(t)+f_2(t)+f_3(t)+\cdots+f_N(t)\over N}

Upload a plot of F_N(t) when N=100.
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Describe the important differences between F_N(t) and f(t). Also describe the important differences between f_N(t) and F_N(t).

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2c. Compare the Fourier series coefficients for f_N(t) with those for F_N(t). Note that the overshoot associated with Gibb's phenomenon is smaller in F_N(t) than in f_N(t); what might explain this difference?