Power Series

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Power Series
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So?
So, start asking yourself this question Can x be
0, 1/2 , 1/2, 3/4, 3/4, 1, 1, 3/2, 3/2, 2,
2, and so on?
3
Again, ask yourself this question Can x be 0,
1/2 , 1/2, 3/4, 3/4, 1, 1, 3/2, 3/2, 2, 2,
and so on?
4
Once more, ask yourself this question Can x be
0, 1/2 , 1/2, 3/4, 3/4, 1, 1, 3/2, 3/2, 2,
2, and so on?
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1, 4)
½(3) 1.5
(2, 2
½(4) 2
½
0
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One number, in particular, will obviously make
the series converge. What is this number? Hint
this number will make every term 0.
x 3
Of course, x may be other numbers too. If so, on
the number line, we begin at 3 and move to the
right and left to obtain our interval of
convergence.
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Therefore, the IOC is 2, 4), and ROC 1.
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What is the number that makes every term 0?
x 1
Again, on the number line, we begin at 1 and move
to the right and left to obtain our interval of
convergence.
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Therefore, the IOC is 0, 2, and ROC 1.
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Answer Kind of like doing the Ratio Test
and
x 3
x 3 lt 1
Step 3 Set above limit lt 1 and solve for x
1 lt x 3 lt 1
3 3 3
2 lt x lt 4
Step 4 Check whether endpoints work or not
?IOC is 2, 4), and ROC 1.
9
Recap
2) We then check the endpoints (to see whether
they will make the power series converges or
not) by substituting each endpoint into the
series. (Step 4)