Consider the alternating series f(x) = x − x^{2} + x^{4} − x^{8} + ... + (−1)^{n }x^{(2n)} + ... , which converges for |x| < 1. Does the limit of f(x) as x approaches 1 from below exist, and if so what is it?

Firstly, note that f(x) = x − f(x^{2}).

If f(x) has limit L as x approaches 1 from below, then L = 1 − L, so that L = ½.

(That is, if the limit exists, then for any e > 0, there exists X such that for all x > X (with |x| < 1), |f(x) − ½| < e.)

We will show that the limit cannot be equal to ½, and hence the limit does not exist.

From here there are a number of routes we could take. We might try to prove that the limit exists, perhaps using one of the standard tests for convergence. Any such attempt would fail. Or we might take a more computational approach, by calculating f(x) for various values of x close to 1. By itself, this approach will not yield a proof, but it may provide some useful hints.

Note that, by the Comparison Test, f(x) is absolutely convergent for 0 x < 1: consider

g(x) = x + x^{2} + x^{4} + x^{8} + ... + x^{(2n)} + ... , and

h(x) = x + x^{2} + x^{3} + x^{4} + ... + x^{n} + ... ,

where h(x), a geometric series, is known to converge.^{ }

Hence we may bracket the terms of f(x), and write f(x) = (x − x^{2}) + (x^{4} − x^{8}) + (x^{16} − x^{32}) + ...

Note that all of the bracketed terms are positive, as 0 < a < b x^{a} > x^{b}, for 0 < x < 1.

This allows us to easily calculate a lower bound for f(x), for a given value of x.

The table below shows the sum of 16 terms (or eight bracketed terms) of f(x) for x = 0, 1/2, 3/4, 7/8, ... , 4095/4096, correct to six decimal places. Notice that f(4095/4096) > ½. We can be sure of this because we are taking the sum of a finite number of positive terms.

m | x = 1 − (½)^{m} | f(x): sum of 16 terms |
---|---|---|

0 | 0 | 0 |

1 | 1/2 | 0.308609 |

2 | 3/4 | 0.413715 |

3 | 7/8 | 0.456269 |

4 | 15/16 | 0.479453 |

5 | 31/32 | 0.489163 |

6 | 63/64 | 0.495031 |

7 | 127/128 | 0.497185 |

8 | 255/256 | 0.498879 |

9 | 511/512 | 0.499179 |

10 | 1023/1024 | 0.499838 |

11 | 2047/2048 | 0.499676 |

12 | 4095/4096 | 0.500078 |

Next, note that f(x) = x − x^{2} + f(x^{4}).

Since x > x^{2} for 0 < x < 1, we have f(x) > f(x^{4}).

Putting these results together, and setting a = 4095/4096, we find that:

½ < f(a) < f(a^{1/4}) < f(a^{1/16}) < f(a^{1/64}) < ... .

The sequence {a, a^{1/4}, a^{1/16}, a^{1/64}, ... } is strictly increasing, and approaches 1.

This shows explicitly that, if we take e = 0.00007, there is *no* value of X such that for all x > X, (with |x| < 1), |f(x) − ½| < e. (No matter what value of X we choose, we can always find an element p > X of the sequence {a, a^{1/4}, a^{1/16}, a^{1/64}, ... }, for which f(p) − ½ > e.)

Hence f(x) does not approach ½, and thus has no limit at all.^{ }

Therefore, the limit of f(x) as x approaches 1 from below does *not* exist.

It may be asked what f(x) does as x approaches 1. The answer is that it oscillates infinitely many times, with each oscillation about four times quicker than the previous one. See the solution given in the reference below for more details.

Source: Puzzle 8 on Noam's Mathematical Miscellany