smc-1189-50-X-topic Series

Consider the geometric series `1-tan^2 theta + tan^4 theta- …`

When the limiting sum exists, find its value in simplest form. (2 marks)

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For what values of `theta` in the interval
`-pi/2 < theta < pi/2` does the limiting sum of the series exist? (2 marks)

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Show Answers Only

a. `cos^2 theta`

b. `-pi/4 < theta < pi/4`

Show Worked Solution

a. `1-tan^2 theta + tan^4 theta- …`

`=>\ text(GP where)\ \ a=1,\ \ r=a_2/a_1= -tan^2 theta`

`S_oo= 1/(1-(-tan^2 theta))= 1/(1 + tan^2 theta)= 1/(sec^2 theta)= cos^2 theta`

b. `text(Find)\ theta\ text(such that)\ |r |<1:`