# Using Series Approximations

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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:

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):

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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):

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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:

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Formatting Help

**2b.**
Let F_N(t) represent the average of the first N finite sums from part 2a.

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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?