Tell Which of the Following Three Series Cannot Be Found $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$

question 179

Multiple Choice

Tell which of the following three series cannot be found convergent by the Ratio Test but can be found convergent by comparison with a p-series.
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 + 3 ^ { n } }$
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { n ^ { 3 } + 4 }$
3) $\sum _ { n = 1 } ^ { \infty } \frac { \sqrt { n } } { n ^ { 2 } + n }$

• A)None
• B)1
• C)2
• D)3
• E)1, 2
• F)1, 3
• G)2, 3
• H)1, 2, 3

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Definitions:

Cylinders

Three-dimensional geometric shapes with straight parallel sides and a circular or oval cross-section.

Drill Bits

Tools made of hard material, often high-speed steel or carbide, used to create cylindrical holes.

Chamfering

The process of creating a beveled edge at the intersection of two surfaces to remove sharp edges for safety, aesthetics, or part assembly.

Groove

A long, narrow cut or depression in a hard material, often used to fit another component or allow the passage of fluids.