MAT 1236 Calculus III Section 11.2 Series Part II

HW… WebAssign 11.2 Part II Quiz: 11.2 II

Part II Introduce Geometric Series, Harmonic Series Test for Divergence

Standard Series #1 Geometric Series (G.S.)

If |r|<1, then If |r|  1, then is divergent

Proof:

PPFTNE State and prove the convergence of the geometric series.

Example 3

Please pay attention to the important details of the solutions Identify the series as G.S. with the parameters a, and r From the absolute value of r, conclude that the series is convergent or divergent (Determine the sum if it is required)

Example 4

Example 5 Find the value of x for which is convergent

Standard Series #2 Harmonic Series The harmonic series is divergent Note: It is not intuitively obvious that the harmonic series is divergent. Proof: (skip)

Theorem If is convergent then Why?

Theorem If is convergent then

Theorem If is convergent then

Theorem If is convergent then

Test For Divergence If, then is divergent

Example 6 By the Test for Divergence,…

Example 6

PPFTNE T or F? If, then is convergent

Theorem