Determine If the Series Converges or Diverges $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n + 1 } } - \frac { 1 } { \sqrt { n + 2 } } \right)$$

question 321

Multiple Choice

Determine if the series converges or diverges. If the series converges, find its sum.
- $$\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { \sqrt { n + 1 } } - \frac { 1 } { \sqrt { n + 2 } } \right)$$

• A) converges; \(\frac { 1 } { \sqrt { 6 } }\)
• B) converges; \(\frac { 1 } { \sqrt { 2 } }\)
• C) converges; \(\frac { 1 } { \sqrt { 3 } }\)
• D) diverges

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Definitions:

Pleural Cavity

The thin fluid-filled space between the two pleural membranes surrounding each lung, providing lubrication during breathing.

Pleura

The thin, protective membranes surrounding the lungs and lining the inner chest wall, facilitating smooth movement of the lungs during breathing.

Pleurisy

An inflammation of the pleurae, which are the layers of tissue lining the lungs and inner chest wall, often causing sharp chest pain with breathing.

Pleural Cavity

The thin fluid-filled space between the two layers of the pleura surrounding the lungs, providing lubrication during respiratory movements.