Power Series

Definition:  A  Power Series  is a series of the form 
(∑_(n=0)^∞)(Subscript[c, n])(x^n) = (Subscript[c,  0]) + (Subscript[c, 1])x + (Subscript[c,  2])(x^2) + (Subscript[c,  3])(x^3) + .... 
where x is a variable, the(Subscript[c, n])’s   are constants called the coefficients  of the series.

(∑_(n=0)^∞)(x^n)

x = 0, x = (1/4)

When x = 0, this power series is (∑_(n=0)^∞)(0^n) = 1 +  0 + 0 + .... . The partial sums are all 1  and hence converge to 1.
When x = (1/4), this power series is (∑_(n=0)^∞)((1^n)/(4^n)) =  (∑_(n=0)^∞)(1/(4^n))  . We recognize this as a geometric series with r  = (1/4) < 1 which converges.
The sequence of partial sums at x = (1/4) are shown in the animation below.

Manipulate[ListPlot[Table[{1/4,(∑_(k=0)^n)(1/4)^k},{n,1,M}],  PlotRange->{-.1,1.5},PlotStyle->{Red,PointSize[Small]},AxesLabel->{x,y},PlotLabel->"points (1/4, (∑_(k = 0)^M)(1/4)^k), M = 1...100"],{M,1,100,1}]

x = 2

When x = 2, this power series is (∑_(n=0)^∞)(2^n) . We recognize this as a geometric series with r  = 2 > 1 which diverges.
The sequence of partial sums at x = 2 are shown in the animation below.

Manipulate[ListPlot[Table[{2,(∑_(k=0)^n)2^k},{n,1,M}],  PlotRange->{-.1,1026},PlotStyle->{Red,PointSize[Small]},AxesLabel->{x,y},PlotLabel->"points (2, (∑_(k = 0)^M)(2)^k), M = 1...20"],{M,1,10,1}]

Partial Sums of the Power Series

Since (∑_(n=0)^∞)(x^n) is a power series with r = x, we can conclude that this power series converges when |x| < 1 and diverges otherwise.

Note that the partial sums for the power series are polynomials. 
 (∑_(n=0)^1)(x^n) = 1 + x
 (∑_(n=0)^2)(x^n) = 1 + x + (x^2)
  (∑_(n=0)^3)(x^n) = 1 + x + (x^2)+ (x^3)
   (∑_(n=0)^4)(x^n) = 1 + x + (x^2)+ (x^3) +(x^4)
   etc....
We can see what happens the  partial sums  (∑_(n=0)^M)(x^n) for all values of x by examining the graphs of these polynomials.
Below we show an animation of the first twenty partial sums. We see  that as M increase the graph of the partial sum gets more stable on the interval -1 < x < 1 reflecting the fact that the the series converges for values of x in this interval.

Manipulate[Plot[(∑_(k=0)^M)((x)^k), {x, -3, 5}, PlotRange->{-5,5}],{M, 1, 20,1}]

In fact, we  see that the graphs of the partial sums converge to the graph of the function f(x) = (1/(1 - x)) 
for -1 < x < 1.

Manipulate[Plot[{(1/(1-(x))),(∑_(k=0)^M)(x)^k}, {x, -3, 5}, PlotRange->{-5,50}, PlotStyle->{Red,Blue,Thick}],{M, 5, 50,5}]

(∑_(n=0)^∞)((((x/2)))^n)

x = 1

When x = 1, this power series is (∑_(n=0)^∞)((1^n)/(2^n)) =  (∑_(n=0)^∞)(1/(2^n))  . We recognize this as a geometric series with r  = (1/2) < 1 which converges.
The sequence of partial sums at x = 1 are shown in the animation below.

Manipulate[ListPlot[Table[{1,(∑_(k=0)^n)(1/2)^k},{n,1,M}],  PlotRange->{-.1,2.5},PlotStyle->{Red,PointSize[Small]},AxesLabel->{x,y},PlotLabel->"points (1, (∑_(k = 0)^M)(1/2)^k), M = 1...100"],{M,1,100,1}]

x = 2

When x = 2, this power series is (∑_(n=0)^∞)((2^n)/(2^n)) =  (∑_(n=0)^∞)(1^n)  . We recognize this as a geometric series with r  = 1 which diverges.
The sequence of partial sums at x = 2 are shown in the animation below.

Manipulate[ListPlot[Table[{2,(∑_(k=0)^n)1^k},{n,1,M}],  PlotRange->{-.1,20},PlotStyle->{Red,PointSize[Small]},AxesLabel->{x,y},PlotLabel->"points (2, (∑_(k = 0)^M)(1)^k), M = 1...20"],{M,1,20,1}]

Partial Sums of the Power Series

Since (∑_(n=0)^∞)((x^n)/(2^n)) is a power series with r = (x/2), we can conclude that this power series converges when |(x/2)| < 1 and diverges otherwise.

The partial sums for the power series are polynomials. 
 (∑_(n=0)^1)((x^n)/(2^n)) = 1 + (x/2)
 (∑_(n=0)^2)((x^n)/(2^n)) = 1 + (x/2) + ((x^2)/(2^2))
  (∑_(n=0)^3)((x^n)/(2^n)) = 1 + (x/2) + ((x^2)/(2^2))+ ((x^3)/(2^3))
   (∑_(n=0)^4)((x^n)/(2^n)) = 1 + (x/2) + ((x^2)/(2^2))+ ((x^3)/(2^3)) + ((x^4)/(2^4))
   etc....
We can see what happens the  partial sums  (∑_(n=0)^M)((x^n)/(2^n))  for all values of x by examining the graphs of these polynomials.
Below we show an animation of the first twenty partial sums. We see  that as M increase the graph of the partial sum gets more stable on the interval -2 < x < 2 reflecting the fact that the the series converges for values of x in this interval.

Manipulate[Plot[(∑_(k=0)^M)(((x/2))^k), {x, -3, 5}, PlotRange->{-5,5}],{M, 1, 20,1}]

In fact, we  see that the graphs of the partial sums converge to the graph of the function f(x) = (1/(1 - (x/2))) 
for -2 < x < 2.

Manipulate[Plot[{(1/(1-((x/2)))),(∑_(k=0)^M)((x/2))^k}, {x, -3, 5}, PlotRange->{-5,50}, PlotStyle->{Red,Blue,Thick}],{M, 5, 50,5}]

(∑_(n=0)^∞)((((x))^n)/(n !))

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging (to the values on the red line) on the interval (-∞, ∞). We see that the computer has difficulty coping
with the calculations for x < -40.

Manipulate[Plot[{Exp[x],(∑_(k=0)^M)(x)^k/k!} ,{x, -70, 20}, PlotRange->{-20,100},PlotStyle->{Red,Blue,Thick}],{M, 5, 200,5}]

(∑_(n=0)^∞)((((x+2))^n)/((n + 1)(4^n)))

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging (to the values on the red line) on the interval [-6, 2).

Manipulate[Plot[{-((4 Log[((2-x)/4)])/(2+x)),(∑_(k=0)^n)(x+2)^k/((k+1) 4^k)} ,{x, -10, 10}, PlotRange->{-20,500},PlotStyle->{Red,Blue,Thick}],{n, 5, 200,5}]

(∑_(n=0)^∞)((((x-1))^n)/((((n + 1))^3)(3^n)))

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging (to the values on the red line) on the interval [-2, 4].

Manipulate[Plot[{((3 PolyLog[3,(1/3) (-1+x)])/(-1+x)),(∑_(k=0)^M)((1/(k+1)^3)((x-1)/3)^k)} ,{x, -6, 6}, PlotRange->{-2,2},PlotStyle->{Red,Blue,Thick}],{M, 5, 50,5}]

(∑_(n=0)^∞)((((-x))^n)/((2n + 1)))

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging (to the values on the red line) on the interval (-1, 1].

Manipulate[Plot[{1-(x/(2 (((1+x))^2))),(∑_(k=0)^M)(-x)^k/2k+1} ,{x, -2, 2}, PlotRange->{-70,70},PlotStyle->{Red,Blue,Thick}],{M, 1, 50,3}]

(∑_(n=0)^∞)(((((-1))^((n)))(((x))^((2n+1))))/((2n + 1))) = ArcTan[x],  if  x in [-1, 1).

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging to ArcTan[x] ( the red line) on the interval (-1, 1].

Manipulate[Plot[{ArcTan[x],(∑_(k=0)^M)((-1)^k x^(2k+1))/(2k+1)} ,{x, -10, 10}, PlotRange->{-4,4},PlotStyle->{Red,Blue,Thick}],{M, 1, 50,3}]

(∑_(n=0)^∞)(((((-1))^((n)))(((x))^((n+1))))/((n + 1))) = Ln[1 + x],  if  x in (-1, 1].

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging to Ln[1 + 
x] ( the red line) on the interval (-1, 1].

Manipulate[Plot[{Log[1+x],(∑_(k=0)^M)((-1)^k x^(k+1))/(k+1)} ,{x, -10, 10}, PlotRange->{-4,4},PlotStyle->{Red,Blue,Thick}],{M, 1, 50,1}]

(∑_(n=0)^∞)(((-1))^((n)))(((x))^((7n))) = (1/(1  + (x^7))),  if  x in (-1, 1).

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging to (1/(1+(x^7)))
 ( the red line) on the interval (-1, 1).

Manipulate[Plot[{(1/(1+(x^7))),(∑_(k=0)^M)((-1)^k x^(7k))} ,{x, -10, 10}, PlotRange->{-4,4},PlotStyle->{Red,Blue,Thick}],{M, 1, 50,3}]

(∑_(n=0)^∞)(((-1))^((n)))((((x))^((7n + 1)))/((7n + 1)))+C = ∫(1/(1  + (x^7)))dx,  if  x in (-1, 1).

Partial Sums of the Power Series

The Mth partial sums of this power series are shown below, 
converging to
 (1/(1+(x^7))) =  (2/7) ArcTan[Sec[(π/14)] (x-Sin[(π/14)])] Cos[(π/14)]+(2/7) ArcTan[Sec[((3 π)/14)] (x+Sin[((3 π)/14)])] Cos[((3 π)/14)]+(1/7) Log[1+x]-(1/7) Cos[(π/7)] Log[1+(x^2)-2 x Cos[(π/7)]]-(1/7) Log[1+(x^2)-2 x Sin[(π/14)]] Sin[(π/14)]+(2/7) ArcTan[(x-Cos[(π/7)]) Csc[(π/7)]] Sin[(π/7)]+(1/7) Log[1+(x^2)+2 x Sin[((3 π)/14)]] Sin[((3 π)/14)] +(1/49) π Cos[(π/14)]-(3/49) π Cos[((3 π)/14)]+(5/49) π Sin[(π/7)]
 ( the red line) on the interval (-1, 1).

Manipulate[Plot[{ (2/7) ArcTan[Sec[(π/14)] (x-Sin[(π/14)])] Cos[(π/14)]+(2/7) ArcTan[Sec[((3 π)/14)] (x+Sin[((3 π)/14)])] Cos[((3 π)/14)]+(1/7) Log[1+x]-(1/7) Cos[(π/7)] Log[1+(x^2)-2 x Cos[(π/7)]]-(1/7) Log[1+(x^2)-2 x Sin[(π/14)]] Sin[(π/14)]+(2/7) ArcTan[(x-Cos[(π/7)]) Csc[(π/7)]] Sin[(π/7)]+(1/7) Log[1+(x^2)+2 x Sin[((3 π)/14)]] Sin[((3 π)/14)] +(1/49) π Cos[(π/14)]-(3/49) π Cos[((3 π)/14)]+(5/49) π Sin[(π/7)],(∑_(k=0)^M)((-1)^k x^(7k+1)/(7k+1))} ,{x, -10, 10}, PlotRange->{-4,4},PlotStyle->{Red,Blue,Thick}],{M, 1, 50,1}]