Activity 8.5.7.
(a)
Show that the Taylor series centered at \(0\) for \(\cos(x)\) converges to \(\cos(x)\) for every real number \(x\text{.}\)
(b)
Next we consider the Taylor series for \(e^x\text{.}\)
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Show that the Taylor series centered at \(0\) for \(e^x\) converges to \(e^x\) for every nonnegative value of \(x\text{.}\)
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Show that the Taylor series centered at \(0\) for \(e^x\) converges to \(e^x\) for every negative value of \(x\text{.}\)
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Explain why the Taylor series centered at \(0\) for \(e^x\) converges to \(e^x\) for every real number \(x\text{.}\) Recall that we earlier showed that the Taylor series centered at \(0\) for \(e^x\) converges for all \(x\text{,}\) and we have now completed the argument that the Taylor series for \(e^x\) actually converges to \(e^x\) for all \(x\text{.}\)
(c)
Let \(P_n(x)\) be the \(n\)th order Taylor polynomial for \(e^x\) centered at \(0\text{.}\) Find a value of \(n\) so that \(P_n(5)\) approximates \(e^5\) correct to \(8\) decimal places.
