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Explain the concepts first, and then step by step explanations for all parts with clear presentation, thank you
Explain the concepts first, and then step by step explanations for all parts with clear presentation, thank you.
We can use power series to define functions. Thus, for example, we can try to
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define the function f viabut the definition of 2' only makes sense when the series converges. Find the radius of convergence and
interval of convergence for this power series. Write down also the sum of the series for X inside the radius of
convergence.
Expert Solution
f( x 1 = 5 2 x" = 2 2 (2)" - an By Ratbtust Lz lim ant , an z the servies converge is Let - (2) c1 9 121 5 3 3' - 3 < x < 3 If x =-3, 2 (-1)" diverge by Alternating series test 5 2XI diverge by divergence series test of x = 3 , - 32 x C3 - 2 Interval of convergence is i-e (-313 ) Radiny of convergence is R = 3 - (-3) = 3 2 and series Sum 2 5 = 2 6 1 + ( 3 ) + ( 2 ) 2 + - - ] = 2 IX 1 1 - 2 2 2 X3 6 with 3 - X 3 -X 2 121 <3 ". By grometric Dorey a tar tan't - converge to 9 with 191 1
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