Can removing brackets destroy diverging to +infinity of a series?

Question

I want to find an example such that after inserting parentheses in initial series, new series diverges to $+\infty$ but original series is not diverging to $+\infty$ (if such example exists).

Motivation: there are examples that show how adding parentheses can make a series converging, for example $$1-1+1-1+1-1+\dots $$ does not converge, but $$(1-1)+(1-1)+(1-1)+\dots $$ converges. Can removing parentheses also have such an effect?

Answer 1

I suppose you are looking for some constuction like this: Start with any two sequences $a_n$ and $b_n$ and let $c_1=a_1$, $c_{2n}=b_n-c_{2n-1}$, $c_{2n+1}=a_n-c_{2n}$. Then $$ c_1+(c_2+c_3)+(c_4+c_5)+\ldots=a_1+a_2+a_3+\ldots$$ $$ (c_1+c_2)+(c_3+c_4)+(c_5+c_6)+\ldots=b_1+b_2+b_3+\ldots$$

Answer 2

Take a symmetric random walk $S_n = \sum_{k=1}^n X_k$ where $X_k$ are i.i.d and $P(X_1 = 1) = P(X_1 = -1) = \frac{1}{2}$.

It's well known that $\limsup S_n = +\infty$ and $\liminf S_n = -\infty$.

So if we rebracket $S_n$ to approach its $\limsup$, we get an example

Answer 3

No.

If you group consecutive terms of a series with parentheses to produce a new series, the partial sums of the new series will be a subsequence of the partial sums of the old series. Therefore, if the old series converges, its sum must also be the sum of the new series.

(note that, for most arguments involving convergence, including the one above, it is more appropriate to think of sequence with limit $+\infty$ as converging to $+\infty$ rather than being a divergent sequence)