The open interval of convergence is 1 then attached image.
We now approach series from a different perspective: as a function. Given a value of x , we evaluate f(x) by finding the sum of a particular series that depends on x (assuming the series converges). We start this new approach to series with a definition.
A series can be represented in a compact form, called summation or sigma notation. The Greek capital letter, ∑ , is used to represent the sum.
Let {[tex]a_{n}[/tex]} be a sequence, let x be a variable, and let c be a real number.
∑(n=0 to ∞) [tex]a_{n} x^{n}[/tex] = [tex]a_{0} +a_{1} x+a_{2} x^{2} +.....[/tex]
∑(n=0 to ∞) [tex]a_{n} (x-c)^{n}[/tex] = [tex]a_{0} +a_{1} (x-c)+a_{2} (x-c)^{2} +.....[/tex]
One of the conventions we adopt is that [tex]x^{0}[/tex] = 1 regardless of the value of x . Therefore
∑(n=0 to ∞) [tex]x^{n}[/tex] = [tex]1+x+x^{2} +x^{3} +......[/tex]
List of Integral Formulas
∫ 1 dx = x + C.
∫ a dx = ax+ C.
∫ [tex]x^{n}[/tex] dx = (([tex]x^{n+1}[/tex])/(n+1))+C ; n≠1.
∫ sin x dx = – cos x + C.
∫ cos x dx = sin x + C.
Therefore,
The open interval of convergence is 1 then attached image.
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