Mathematics

Mathematics

Problem

Which of the following statements correctly characterizes the convergence of the series ∑n=1∞25n\displaystyle\sum_{n=1}^{\infty}\dfrac{2}{5^{n}} to the value 12\dfrac{1}{2}, where SnS_n denotes its nnth partial sum? a) For every N∈NN\in\mathbb{N} there exists ε>0\varepsilon>0 such that if n≥Nn\ge N then ∣Sn−12∣<ε|S_n-\tfrac12|<\varepsilon. b) For every ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that ∣SN−12∣<ε|S_N-\tfrac12|<\varepsilon. c) For every ε>0\varepsilon>0 there exists N∈NN\in\mathbb{N} such that if n≥Nn\ge N then ∣Sn−12∣<ε|S_n-\tfrac12|<\varepsilon. d) There exists N∈NN\in\mathbb{N} such that for all ε>0\varepsilon>0, if n≥Nn\ge N then ∣Sn−12∣<ε|S_n-\tfrac12|<\varepsilon.

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