Homework SourseUse the formula a1r a ar ar2 ar3 valid for absr 1 to exp
Use the formula a/(1-r) = a + ar + ar^2 +ar^3... valid for abs(r) < 1 to express the following function:
f(x) = x/(1-x), as a power series a0 + a1x + a2x^2 +.....+{a(n)}x^n. Give a formula for the coefficient {a(n)} and determine the values in which the series converges
f(x) = x/(1-x), as a power series a0 + a1x + a2x^2 +.....+{a(n)}x^n. Give a formula for the coefficient {a(n)} and determine the values in which the series converges
Solution
Use the formula a/(1-r) = a + ar + ar^2 +ar^3... valid for abs(r) < 1 to express the following function:
f(x) = x/(1-x), as a power series a0 + a1x + a2x^2 +.....+{a(n)}x^n. Give a formula for the coefficient {a(n)} and determine the values in which the series converges
f(x) = x / (1-x)
f(x) = x (1-x)^-1 = x ( 1 +x +x ^2 + x^3...............)
= x + x^2 + x^3 +x^4..........
a(n) = x^n+1
series converges for |x| < 1
