10.6.2 Convergence of the Arctangent Series

Once more, here is the formula for the arctangent series:

tan -1 ( x ) = x - 1 3 ⁢ x 3 + 1 5 ⁢ x 5 - ⋅⋅⋅ = ∑ k = 0 ∞ ( - 1 ) k x 2 k + 1 2 k + 1 , if ⁢   | x | < 1.

The presumed domain in which this series converges, namely, `text[|] x text[|]<1`, was inherited from the geometric series. But do we really know that the series converges for these values of `x`? Do we know that it diverges for other values of `x`? Those are questions we can now answer. You have already partially answered one of these questions: In Section 10.5 you showed that the series converges for `x=1`, which is not in the presumed domain.

Note that the arctangent series is alternating for every value of `x`. That's clear for positive values of `x`; if `x` is negative, so is every odd power of `x`, which means the signs still alternate, but in the opposite pattern, `- + - + cdots`. In fact, wherever the series converges, it defines an odd function, which is no surprise, since arctangent is also an odd function. Since the series for `x=-1` is the negative of the series for `x=1`, you already know that it converges for `x=-1,` apparently to `-pi//4`.

Example 1    Show that the arctangent series converges if `text[|] x text[|]<=1`.

Solution   We already know the series converges if `x=+-1`. If `x` is a number between `-1` and `1`, then the powers of `x` approach zero, so the terms of the series do too. Is each term larger (in absolute value) than the one after it? First we look at the numerators of consecutive terms:

| x | 2 k + 3 = | x | 2 | x | 2 k + 1 < | x | 2 k + 1

because `text[|] x text[|]<1`. We already know that the denominators are increasing, so their reciprocals are decreasing. Thus the absolute value of the `k`th term is the product of two factors, `text[|] x text[|]^(2k+1)` and `1text[/(]2k+1text[)]`, both of which are larger than the corresponding factors in the `text[(]k+1text[)]`th term. Hence the terms really do decrease in size. It follows from the Alternating Series Test that the arctangent series converges for all numbers `x` in the interval `[-1,1]`.

Checkpoint 1