Maclaurin series for e^x and prove represent e^x for all x

Question

Suresh Chand · Accepted Answer

Solution: Let f(x) = ex By Maclaurin expansion we have, f(x) = f(0) + x f'(o) + (frac{x^2}{2!})f''(0) + (frac{x^3}{3!})f'''(0) + ... Here, f(x) = ex Differentiating we get, f'(x) = ex = f''(x) = f'''(x) = ....          ,(forall) x At x = 0, f(0) = e0 = 1 = f'(0) = f''(0) = f'''(0) = ....      ,(forall) x Then (i) becomes (f(x) = 1 + x.1 + frac{x^2}{2!}.1 + frac{x^3}{3!}+ ....) = (1 + x + frac{x^2}{2!}.1 + frac{x^3}{3!}+ ....) This is the Maclaurin's series of ex Also, (f(x) = 1 + x.1 + frac{x^2}{2!}.1 + frac{x^3}{3!}+  ....  + frac{x^n}{n!} + R_n(x)) Where (R_n(x)) is the remainder of the series. Since, (left | R_n(x)) right | leq frac{x^{n+1}}{(n + 1)!)}to 0 as nn to ∞) This shows that ex converges to its Maclaurin's series

Find the Maclaurin series for e^x and prove that it represents e^x for all x.

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Find the Maclaurin series for ex and prove that it represents ex for all x.

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Find the Maclaurin series for e^x and prove that it represents e^x for all x.