Determine Whether the Given Series Is Convergent or Divergent $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } + 1 } { 5 n ^ { 3 } - n + 4 }$

question 199

Short Answer

Determine whether the given series is convergent or divergent. Indicate the test you use and show any necessary computation.(a) $\sum _ { n = 1 } ^ { \infty } \frac { 2 n ^ { 2 } + 1 } { 5 n ^ { 3 } - n + 4 }$ (e) $\sum _ { n = 1 } ^ { \infty } \tan ^ { - 1 } n$ (i) $\sum _ { n = 1 } ^ { \infty } \frac { 4 ^ { n } } { 2 ^ { n } + 3 ^ { n } }$ (b) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 + \sin n } { n } \right) ^ { 2 }$ (f) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n \sqrt { 1 + \ln n } }$ (j) $\sum _ { n = 1 } ^ { \infty } \ln \left( 1 + \frac { 1 } { n } \right)$ (c) $\sum _ { n = 1 } ^ { \infty } n \cdot \sin \left( \frac { 1 } { n } \right)$ (g) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { ( n + 1 ) ^ { 3 } }$ (k) $\sum _ { n = 1 } ^ { \infty } n \cdot e ^ { - n ^ { 2 } }$ (d) $\sum _ { n = 1 } ^ { \infty } \left( \frac { 2 } { n \sqrt { n } } + \frac { 3 } { n ^ { 3 } } \right)$ (h) $\sum _ { n = 1 } ^ { \infty } \frac { \ln n } { n }$

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Definitions:

Average Product

The output per unit of input, calculated by dividing total production by the number of units of input.

Marginal Product

The additional output that is produced by adding one more unit of a particular input, keeping other inputs constant.

Short-Run Curve

Refers to a period in economics during which at least one factor of production is fixed, affecting production and cost decisions.

Marginal Product

The change in output resulting from employing one more unit of a particular input while holding all other inputs constant.