Find the Maclaurin series for ex and prove that it represents ex for a | Mathematics I 2074

Mathematics I207410 marksNumerical: Maclaurin series, IVP, sphere volumeTaylor's and Maclaurin's series

Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all x. Define initial value problem. Solve that initial value problem of $y' + 5y = 1$ , $y(0) = 2$ . Find the volume of a sphere of radius r. [4+4+2]

Find the Maclaurin series for $e^x$ and prove that it represents $e^x$ for all x. Define initial value problem. Solve that initial value problem of $y' + 5y = 1$, $y(0) = 2$. Find the volume of a sphere of radius r. [4+4+2]

(a) Maclaurin Series for $e^x$ and Proof The Maclaurin series of a function is: $f(x) = \sum{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}\,x^n$ For $f(x) = e^x$ , all derivatives equal $e^x$ : $f^{(n)}(x) = e^x \implies f^{(n)}(0) = 1 \quad \text{for all } n$ Therefore: $$\boxed{e^x = \sum{n=0}^{\infty}...

(a) Maclaurin Series for $e^x$ and Proof The Maclaurin series of a function is: $$f(x) = \sum{n=0}^{\infty} \frac{f^{(n)}(0)}{n!},x^n$$ For $f(x) = e^x$, all derivatives equal $e^x$: $$f^{(n)}(x) = e^x \implies f^{(n)}(0) = 1 \quad \text{for all } n$$ Therefore: $$\boxed{e^x = \sum{n=0}^{\infty}...

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Find the Maclaurin series for ex and prove that it represents ex for all x. Define initial value problem. Solve that initial value problem of y' + 5y = 1, y(0) = 2. Find the volume of a sphere of radius r. [4+4+2]