Macaurin series of 10^x, please

Question

anonymous · Answer

10*10*10*10*10*10*10*10*10*10

anonymous · Answer

let f(x)=10^x
$$f \prime \left( x \right)=10^x \log 10$$
$$f \prime \prime \left( x \right)=10^x \left( \log 10 \right)^2$$
$$f \prime \prime \prime \left( x \right)=10^x \left( \log10 \right)^3$$
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$$f^n \left( x \right)=10^x \left( \log 10 \right)^{n}$$
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$$f \left( 0 \right)=10^0=1,f \prime \left( 0 \right)=10^0 (\log 10)=\log10$$
$$f \prime \prime \left( 0 \right)=10^0 \left( \log 10 \right)^2$$
$$f \prime \prime \prime \left( 0 \right)=10^0\left( \log 10 \right)^3$$
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$$f^n \left( 0 \right)=10^0\left( \log10 \right)^n$$
Maclaurin series for f(x)
$$f(x)=f(0)+x f \prime \left( 0 \right)+\frac{ x^2 }{ 2! }f \prime \prime \left( 0 \right)+\frac{ x^3 }{ 3! }f \prime \prime \prime \left( 0 \right) +...+\frac{ x^n }{ n! }f^n \left( 0 \right) +...$$
put the values and complete it.

anonymous · Answer

Alternatively, you can get it from the power series for $e^x$:
$$10^x=e^{\ln 10^x}=e^{x\ln10}$$
$$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}~~\implies~~10^x=e^{x\ln10}=\sum_{n=0}^\infty \frac{(x\ln10)^n}{n!}$$

thesmartone · Answer

Welcome to OS!!