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kondor19780726 [428]
3 years ago
6

EXAMPLE 2 Prove that 9ex is equal to the sum of its Maclaurin series. SOLUTION If f(x) = 9ex, then f (n + 1)(x) = for all n. If

d is any positive number and |x| ≤ d, then |f (n + 1)(x)| = ≤ 9ed. So Taylor's Inequality, with a = 0 and M = 9ed, says that |Rn(x)| ≤ (n + 1)! |x|n + 1 for |x| ≤ d. Notice that the same constant M = 9ed works for every value of n. But, from this equation, we have lim n → [infinity] 9ed (n + 1)! |x|n + 1 = 9ed lim n → [infinity] |x|n + 1 (n + 1)! = . It follows from the Squeeze Theorem that lim n → [infinity] |Rn(x)| = 0 and therefore lim n → [infinity] Rn(x) = for all values of x. By this theorem, 9ex is equal to the sum of its Maclaurin series, that is, 9ex = [infinity] 9xn n! n = 0 for all x.
Mathematics
1 answer:
amm18123 years ago
6 0

Answer:

To Prove: 9e^x is equal to the sum of its Maclaurin series.

Step-by-step explanation:

If f(x) = 9e^x, then f ^{(n + 1)(x)} =9e^x for all n. If d is any positive number and   |x| ≤ d, then |f^{(n + 1)(x)}| = 9e^x\leq  9e^d.

So Taylor's Inequality, with a = 0 and M = 9e^d, says that |R_n(x)| \leq \dfrac{9e^d}{(n+1)!} |x|^{n + 1} \:for\: |x| \leq  d.

Notice that the same constant M = 9e^d works for every value of n.

But, since lim_{n\to\infty}\dfrac{x^n}{n!} =0 $ for every real number x$,

We have lim_{n\to\infty} \dfrac{9e^d}{(n+1)!} |x|^{n + 1} =9e^d lim_{n\to\infty} \dfrac{|x|^{n + 1}}{(n+1)!} =0

It follows from the Squeeze Theorem that lim_{n\to\infty} |R_n(x)|=0 and therefore lim_{n\to\infty} R_n(x)=0 for all values of x.

THEOREM\\If f(x)=T_n(x)+R_n(x), $where $T_n $is the nth degree Taylor Polynomial of f at a and  $ lim_{n\to\infty} R_n(x)=0 \:  for \: |x-a|

By this theorem above, 9e^x is equal to the sum of its Maclaurin series, that is,

9e^x=\sum_{n=0}^{\infty}\frac{9x^n}{n!}  for all x.

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