Chapter 19: Problem 2

Use the power series expansion of $\mathrm{e}^{x}$ to show that $$ \mathrm{e}^{2 x}=1+2 x+2 x^{2}+\frac{4 x^{3}}{3}+\cdots $$

Short Answer

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Question: Write the power series expansion of e^(2x) in terms of x. Answer: The power series expansion of e^(2x) is given by: $$ e^{2x} = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \cdots $$

Step by step solution

Write down the power series expansion of e^x #

We know that the power series expansion of e^(x) is given by $$ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots $$ This series converges for all values of x.

Apply the substitution x -> 2x #

To find the power series expansion of e^(2x), we substitute 2x in place of x in the power series expansion of e^x. We get $$ e^{2x} = 1 + (2x) + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \cdots $$

Simplify the terms #

Now we simplify the terms in the series: $$ e^{2x} = 1 + 2x + \frac{4x^2}{2} + \frac{8x^3}{6} + \cdots $$

Simplify the terms with their constant factors #

Simplify the series after canceling out the constant factors in each term: $$ e^{2x} = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \cdots $$

Write out the final answer #

We have derived the power series expansion of e^(2x) in terms of x as required: $$ e^{2x} = 1 + 2x + 2x^2 + \frac{4x^3}{3} + \cdots $$

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Use the power series expansion of \(\mathrm{e}^{x}\) to show that $$ \mathrm{e}^{2 x}=1+2 x+2 x^{2}+\frac{4 x^{3}}{3}+\cdots $$

Short Answer

Step by step solution

Write down the power series expansion of e^x #

Apply the substitution x -> 2x #

Simplify the terms #

Simplify the terms with their constant factors #

Write out the final answer #

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