Solution Found!
(For those whove thought about convergence issues) Check that the power series expansion
Chapter 7, Problem 15(choose chapter or problem)
(For those whove thought about convergence issues) Check that the power series expansion f (x) = _ k=0 xk k! converges for any real number x and that f (x) = ex , as follows. a. Fix x _= 0 and choose an integer K so that K 2|x|. Then show that for k > K, we have |x|k k! C _ 12 _ kK, where C = |x| K . . . |x| 2 |x| 1 is a fixed constant. b. Conclude that the series k=K+1 |x|k k! is bounded by the convergent geometric series C j=1 1 2j and therefore converges and, thus, that the entire original series converges absolutely. c. It is a fact that every convergent power series may be differentiated (on its interval of convergence) term by term to obtain the power series of the derivative (see Spivak, Calculus, Chapter 24). Check that f _ (x) = f (x) and deduce that f (x) = ex .
Questions & Answers
QUESTION:
(For those whove thought about convergence issues) Check that the power series expansion f (x) = _ k=0 xk k! converges for any real number x and that f (x) = ex , as follows. a. Fix x _= 0 and choose an integer K so that K 2|x|. Then show that for k > K, we have |x|k k! C _ 12 _ kK, where C = |x| K . . . |x| 2 |x| 1 is a fixed constant. b. Conclude that the series k=K+1 |x|k k! is bounded by the convergent geometric series C j=1 1 2j and therefore converges and, thus, that the entire original series converges absolutely. c. It is a fact that every convergent power series may be differentiated (on its interval of convergence) term by term to obtain the power series of the derivative (see Spivak, Calculus, Chapter 24). Check that f _ (x) = f (x) and deduce that f (x) = ex .
ANSWER:Step 1 of 4
The power series expansion of a convergence function f(x) is,
It converges for any real number x.