Test 3 Review

Section 8.4 - #9 and #12

Note: I only did part (a) and (b) for question 9 because the rest of the parts follow the same basic methodology.

Note: I only did part (a), (d) and (e) for question 12 because the rest of the parts follow the same basic methodology.

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9. Find the coefficient of \(x^{10}\) in the power series of each of these functions.

\[a) \ (1 + x^5 + x^{10} + x^{15} + ...)^3\] \[b) \ (x^3 + x^4 + x^5 + x^6 + x^7 + ...)^3\] \[c) \ (x^4 + x^5 + x^6)(x^3 + x^4 + x^5 + x^6 + x^7)(1 + x + x^2 + x^3 + x^4 + ...)\] \[d) \ (x^2 + x^4 + x^6 + x^8 + ...)(x^3 + x^6 + x^9 + ...)(x^4 + x^8 + x^{12} + ...)\] \[e) \ (1 + x^2 + x^4 + x^6 + x^8 + ...)(1 + x^4 + x^8 + x^{12} + ...)(1 + x^6 + x^{12} + x^{18} + ...)\]

Hint: Finding the pattern for changing the coefficients or exponents when distributing helps save a lot of time. If you know how to distribute and line terms up when adding, problems like this and #10 should be easy.

Answer :: (a) Coefficient for \(x^{10}\) = \(6\). (b) Coefficient for \(x^{10}\) = \(3\).

12. Find the coefficient of \(x^{12}\) in the power series of each of these functions.

\[a) \ 1/(1+3x)\] \[b) \ 1/(1-2x)^2\] \[c) \ 1/(1+x)^8\] \[d) \ 1/(1-4x)^3\] \[e) \ x^3/(1+4x)^2\]

Hint: Use what we know of our “friendly” function from class to put each question in a more familiar form.

Answer :: (a) Coefficient for \(x^{12}\) = \(531 \ 441\).        (d) Coefficient for \(x^{12}\) = \(1526 \ 726 \ 656\).        (e) Coefficient for \(x^{12}\) = \(-2 \ 621 \ 440\).