Explain the expression that sqrt(1+x) is "close" to 1+(1/2)*x for "small" x.
I tried to solved with epsilon-delta:
Let e>0, so there is delta(e)>0 so that
|x|<delta ==> |sqrt(1+x)-{1+(1/2)*x}|<e
Can anybody help me please...?
Looking to all three factors of the Taylor serie
will become positive and negative alternatly, but the absolute value will become lower for higher n.
will become lower for higher n (for small x)
will become lower for higher n.
As you can see, all the three terms becomes lower for higher n. This means that the lower n terms will become dominant. The smaller x is, the more dominant the first terms will be.