Let $a _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }$

question 209

Essay

Let $a _ { n } = \frac { 1 + 3 ^ { n } } { 1 + 4 \cdot 3 ^ { n } }$ .
(a) Find $\lim _ { n \rightarrow \infty } a _ { n }$ .
(b) Is $\sum a _ { n }$ convergent? Justify your answer.

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Definitions:

Regular Cash Dividend

is a fixed amount paid per share to shareholders, typically on a quarterly basis, from a company's earnings.

Residual Dividend Approach

A policy for setting a company's dividend payment, where the dividend paid is based on earnings minus funds the firm retains to finance the equity portion of its capital budget.

Debt-to-Equity Ratio

A ratio reflecting the comparative amounts of debt and equity used to finance a company’s assets.

Investment Needs

The amount of capital required to fund a project, purchase assets, or support operations within a business.