(i) Find an expression for the sum of the series ∑ r=0 ∞ ​ x r . State the interval of convergence for the series. (ii) Hence, USING SIGMA NOTATION, deduce a series for:

Question

(i) Find an expression for the sum of the series ∑r=0∞ ​x^r. State the interval of convergence for the series.

(ii) Hence, USING SIGMA NOTATION, deduce a series for:

(a) 1/1+x^2​ 

(b) arctanx 

(c) 6/π​

Answer 1

(i) It is clear that the sum converges only for |x| < 1 (otherwise, the terms x^r get arbitrarily large for large r).

Let S be the sum, then S(1-x) = 1, which gives us S = 1/(1-x)

(iia) Let's rewrite the formula above as S = 1/(1-y), then to get 1/(1+x^2), we need y = -x^2; so S in sigma notation would be Σ_r=0^\infty(-x^2)^r = Σ_r=0^\infty(-1)^r*x^(2r)

(iib) note that arctanx = the integral of 1/(1+x^2), so the sigma notation for that would be ∫Σ_r=0^\infty(-x)^(2r)dx = Σ_r=0^\infty(-1)^r∫x^(2r)dx = Σ_r=0^\infty(-1)^r*x^(2r+1)/(2r+1)

If you need |x| >= 1, notice how arctanx = pi/2 - arccot(x) = pi/2-arctan(1/x) and arctan(-x) = -arctanx

(iic) Note that arctanx = pi/6 when x = sqrt(3)/3 < 1. So 6/pi = 1/(Σ_r=0^\infty(-1)^r*x^(2r+1)/(2r+1)) where we plug in x = sqrt(3)/3