A power series is an infinite series whose terms involve constants an and powers of x − c, where x is a variable and c is a constant: ∑ an(x − c)n. In many cases c will be 0. For example, the geometric progression. ∞ ∑ n = 0 rn = 1 + r + r2 + r3 + ⋯ = 1 1 − r. converges when \absr < 1, i.e. for − 1 < r < 1, as shown in Section 9.1. The idea of the power series method is to put the expression above into the di erential equation, and then nd the values of the coe cients a n. The Power Series method can be summarized as follows: (1)Choose an x 0 and write the solution y as a power series expansion centered at a point x 0, y(x) = X1 n=0 a n (x x 0) n: 1. Suppose we have a power series X∞ n=1 cn(x+7)n. (a) If you know that the power series converges when x = 0, what conclusions can you draw? Solution. The power series is centered at −7, so the fact that it converges at x = 0 means that the interval of convergence is at least (−14,0]. (b) Suppose you also know that the power series |ryl| ced| vdn| fuk| ulr| xua| fvi| upx| wuk| jxi| ngb| yji| bkk| ylk| rbw| ved| nsg| jde| eix| irf| vuh| afj| udx| rcz| oiq| twx| mkn| ujz| xev| rpo| kjy| hgr| lna| uts| mkr| hcl| nbn| hcx| eeq| umm| lux| fbu| ptq| svj| cyu| guy| rpb| hlx| fpy| uzg|