# Taylor series problem

I have this equation:

960 - 84.60 * ((1-(1+i)^-12)/i) == 0


I simplify ( 1+i)^-12 with a Taylor series ( 1 + x)^a.

but I obtain i == 0.087201167 but the real result should be i == 0.00753 (approximately).

P.S.: My solution

( 1 + i)^-12 == 1 - 12i + 78i^2 + 364i^3


Then:

1 -(1+i)^-12 == 12i - 78i^2 + 364i^3


Then I collect i and I simplify so I obtain:

960 - 84.60 * (12 - 78 i + 364 i^2) == 0


And I obtain:

-55.20 + 6598.80 i - 30794.40i^2 == 0


I apply the Quadratic Formula and I obtain:

i == 0.087201167 and i = 2.055

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## migrated from mathematica.stackexchange.comSep 19 '13 at 3:26

This question came from our site for users of Mathematica.

I don't see an equation. Could you please provide the code you tried? – Sjoerd C. de Vries Sep 18 '13 at 17:40
Stefano, is this a question about how to use the Mathematica software or a question about how to deal with the equation (i.e. math)? – Szabolcs Sep 18 '13 at 17:43
You lost a factor of 'i' in your "collect and simplify" step. [What this has to do with Mathematica is that, had you used it throughout, this would have been an unlikely occurence.] – Daniel Lichtblau Sep 18 '13 at 18:02
@DanielLichtblau You can check these steps are correct until the last line, the result should be i == 0.00872 || i == 0.205566 it is different by a factor 10. Nevertheless approximate solution i == 0.00872 is correct, it just shouldn't be i == 0.00753. – Artes Sep 18 '13 at 18:26
But How you have found i = 0.00872 and i = 0.205566 – Stefano Sep 18 '13 at 18:29

Solve[960 - 84.60*((1 - (1 + i)^-12)/i) == 0, i, Reals]


{{i -> -1.77547}, {i -> 0.00870777}}

Plot[960 - 84.60*((1 - (1 + i)^-12)/i), {i, -2, .1},
PlotRange -> {-.01, 0.01}, Axes -> {True, False}]


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Thank you very much for your answer but I need the Formula in order to add it to Excel. But I don't obtain 0.0087 that is OK! – Stefano Sep 18 '13 at 17:54

Equation

$$960 - 84.6 \frac{ 1 - (1+i)^{-12}}{i} = 0$$

Taylor expansion

$$(1+i)^{-12} \approx -364 i^3+78 i^2-12 i+1$$

Approximate Equation

$$960 - 84.6 \frac{ 364 i^3-78 i^2+12 i }{i} = 0$$

With solutions

$$i = 0.00872 \;,\;\; i = 0.205566$$

So what is your question here?

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