# How to find the Maclaurin series and the radius of convergence for #f(x)=1/(1+x)^2#?

The Maclaurin series is given by

Recall that the McLaurin series is given by

Now let's recall the first few factorials.

We can see the similarity here. Our first few terms are therefore

The radius of converge is given by the ratio test.

Hopefully this helps!

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To find the Maclaurin series for ( f(x) = \frac{1}{(1 + x)^2} ), we can start by expressing it as a geometric series.

[ \frac{1}{(1 + x)^2} = (1 + x)^{-2} = \sum_{n=0}^{\infty} \binom{-2}{n} x^n ]

The Maclaurin series coefficients are given by the formula ( \binom{-2}{n} = \frac{(-2)(-2-1)(-2-2)\ldots(-2-n+1)}{n!} ).

Simplify the expression:

[ \binom{-2}{n} = \frac{(-2)(-3)(-4)\ldots(-n)}{n!} = \frac{(-1)^n(2)(3)(4)\ldots(n)}{n!} ]

This simplifies to:

[ \binom{-2}{n} = (-1)^n \cdot \frac{n!}{(n)!} = (-1)^n \cdot n ]

So, the Maclaurin series for ( f(x) = \frac{1}{(1 + x)^2} ) is:

[ \sum_{n=0}^{\infty} (-1)^n \cdot n \cdot x^n ]

To find the radius of convergence, we can use the ratio test:

[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| ]

where ( a_n = (-1)^n \cdot n \cdot x^n ).

[ \lim_{n \to \infty} \left| \frac{(-1)^{n+1} \cdot (n+1) \cdot x^{n+1}}{(-1)^n \cdot n \cdot x^n} \right| ]

[ = \lim_{n \to \infty} \left| \frac{(n+1) \cdot x}{n} \right| ]

[ = |x| \lim_{n \to \infty} \frac{n+1}{n} ]

[ = |x| ]

Since the series converges for ( |x| < 1 ), the radius of convergence is ( R = 1 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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