Summary: The theorem states that for a series of real numbers that converges but not absolutely, there exists a rearrangement of the series with partial sums converging to given limits. The proof involves constructing sequences and subsequences to show convergence to the desired limits. Another theorem states that if the series converges absolutely, then every rearrangement of the series converges to the same sum.

Summary: This passage presents the proof of a theorem stating that if two infinite series converge, their respective terms can be multiplied to obtain a new convergent series, even without the condition of absolute convergence.

Summary: This blog post highlights the critical role of absolute convergence in mathematical analysis, exploring theorems and providing examples to illustrate its significance in ensuring reliable and consistent operations on infinite series or integrals.

Summary: A power series converges if the absolute value of the variable is less than the radius of convergence, determined by the limit supremum of the coefficients; the sum of a product series can be computed using partial sums and their differences; if the partial sums of a series are bounded and the coefficients are non-increasing and converge to zero, the product series converges; an alternating, absolutely decreasing, and converging-to-zero sequence leads to a convergent series; and if the radius of convergence of a power series is 1, the coefficients are non-increasing and converge to zero, the series converges for all points on the unit circle except possibly at 1.