Final answer:
The series ∑ₖₙ₁[∞]1/((2k−1)(2k+1)) is a telescoping series that converges. Through partial fraction decomposition, it is shown that terms cancel out, leading to only a few terms that determine the sum and prove convergence.
Step-by-step explanation:
The series ∑ₖₙ₁[∞]1/((2k−1)(2k+1)) is known to be a telescoping series, which means it is a series where many terms cancel out when the series is written out in an extended form. The correct option regarding its convergence or divergence is that the series converges.
To determine the convergence of this series, we can perform partial fraction decomposition. The term 1/((2k-1)(2k+1)) can be written as A/(2k-1) + B/(2k+1) for some constants A and B. Solving for A and B gives A=0.5 and B=-0.5. Therefore, the term becomes (1/(2k-1)) - (1/(2k+1)). When the terms of the series are written out, each term cancels with the next one, leading to only a few terms that determine the sum, which proves convergence.
Step by step explanation:
Perform partial fraction decomposition on the term 1/((2k-1)(2k+1)).
Write the expression in the form A/(2k-1) + B/(2k+1) and solve for A and B.
Observe the cancellation of terms when the series is written out, which leads to convergence.