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I ask members here kindly for their assistance. I'm having some confusion over the process of integrating inequalities, in particular for obtaining the series expansion for the exponential function by integration. The text by Backhouse and Holdsworth (Pure Mathematics 2), shows the expansion of the exponential function by integration with the assumption that (x^n)→0 as n→∞ when |x|<1;

The problem i'm having here is the integration step, where does the negative one come from in the middle of the inequality?

Thanks.

Lukka

*Let the variable x lie in the range of values from 0 to c, where c is any positive constant, thus*

0 < x < c

since e^0 = 1,

∴ 1 < (e^x) < e^c

Integrating from 0 to x gives,

x < (e^x - 1) < x(e^c)0 < x < c

since e^0 = 1,

∴ 1 < (e^x) < e^c

Integrating from 0 to x gives,

x < (e^x - 1) < x(e^c)

The problem i'm having here is the integration step, where does the negative one come from in the middle of the inequality?

Thanks.

Lukka

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