namjoonkermit namjoonkermit
  • 23-04-2020
  • Mathematics
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for the series ∑n=0 → ∞ (4ˣ -7)ⁿ, find the interval of convergence and what the series converges to.

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LammettHash
LammettHash LammettHash
  • 23-04-2020

This series is geometric; substitute [tex]y=4^x-7[/tex] to see how.

Then the series converges if

[tex]|4^x-7|<1\implies-1<4^x-7<1\implies6<4^x<8\implies\log_46<x<\log_48[/tex]

Under these conditions, we have

[tex]\displaystyle\sum_{n=0}^\infty(4^x-7)^n=\frac1{1-(4^x-7)}=\frac1{8-4^x}[/tex]

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