Calc 2 Sum of Alternating Geometric Series

[bigbob123]

Homework Statement

Suppose that An (from n = 0 to inf.) = {1/1, 3/1, 3/4, 9/4, 9/16, 27/16, 27/64, 81/64...} where we start with 1 and then alternate between multiplying by 3 and 1/4. Find the sum of An from n = 0 to n = inf.

Relevant Equations

Sn = A0(1-r)/(1-r) iff |r| < 1

A0 = 1
A1 = 3

3(An-1) / 4(An-2) = An

 

[pasmith]

You can show that (A_n)_{n \geq 0} satisfies a second order linear recurrence of the form <br /> A_{n+2} + pA_{n+1} + qA_n = 0, which has general solution <br /> A_n = C\lambda_1^n + D \lambda_2^n where C and D are constants determined by the values of a_0 and a_1 and \lambda_1 and \lambda_2 are the roots of <br /> \lambda^2 + p\lambda + q = 0.<br />

You then have an expression for A_n in closed form and can proceed to determine whether or not \sum_{n=0}^\infty A_n converges.

 

[Dick]

If you look closely you'll notice that your series consists of two interlaced ordinary geometric series.

 

[bigbob123]

You can show that (A_n)_{n \geq 0} satisfies a second order linear recurrence of the form <br /> A_{n+2} + pA_{n+1} + qA_n = 0, which has general solution <br /> A_n = C\lambda_1^n + D \lambda_2^n where C and D are constants determined by the values of a_0 and a_1 and \lambda_1 and \lambda_2 are the roots of <br /> \lambda^2 + p\lambda + q = 0.<br />

You then have an expression for A_n in closed form and can proceed to determine whether or not \sum_{n=0}^\infty A_n converges.

It seems like the problem isn't linear though- An = 3An-2 / 4An-2

 

[RPinPA]

If you look closely you'll notice that your series consists of two interlaced ordinary geometric series.

Which can be rearranged under certain conditions, which you probably ought to first prove.