Why does Negative binomial expansion have infinite terms

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Why does negative binomial expansion have infinite number of terms and not equal to the example given below

Why is $(x+a)^{-2}$ not equal to $\frac{1}{x^2 + a^2 + 2ax}$? ?

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From binomial series, all binomial expansions have an infinite number of terms.

If $\alpha$ is a non-negative integer, after $\alpha+1$ terms the remaining terms are all zero.