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Maclaurin series Validity?

Im currently revising FP2 and am doing Maclaurin series. I was wondering how you work out the validity for a series from its maclaurin series, and whether there was a general rule for how to work it out.

Thankyou

Reply 1

Zacken

Original post by Eux

Radius of convergence. It's Uni material though, so don't expect to understand all of it.

Reply 2

username1421435

Original post by Zacken

Radius of convergence. It's Uni material though, so don't expect to understand all of it.

So does that not mean we have to work out the validity for, say,

arctan(0.5x-2)

ln(3x+5)

Reply 3

Zacken

Original post by Eux

So does that not mean we have to work out the validity for, say,

arctan(0.5x-2)

ln(3x+5)

Those are standard expansions, though. You should be able to derive them from the formula booklet immediately, give me a sec.

Reply 4

Zacken

Original post by Eux

So does that not mean we have to work out the validity for, say,

arctan(0.5x-2)

ln(3x+5)

You know

\ln(1 +x)

is valid for

|x| \leq 1

x \mapsto f(x)

means the validity is restricted to

|f(x)| \leq 1

.

Write

\ln(3x+5) = \ln 5 + \ln \left(1 + \frac{3x}{5}\right)

then the validity is

\left|\frac{3x}{5}\right| \leq 1 \iff |x|\leq \frac{5}{3}

which is your radius of convergence.

P.S: I was doing my standard rounds on math.stack a few minutes earlier. :tongue:

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Maclaurin series Validity?

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