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Convergence test I

Geometric, Harmonic, p andalternatingDetermine whether the seriesconverges

(a)0

1

4

n

n

∞

=

÷∑ (b) 1

0

4 n

n

∞

−

=

∑(c)

2

10

47

n

n

n

+∞

−

=

∑ (d) 1

1

3n

n∞

−

=

∑(e) 3

1

1

n n

∞

=

∑ (f)43

1n

n∞

−

=

∑(g)

52

31

1

n

nn

∞

=

+∑ (h) 12

1

1( 1) n

n n

∞+

=

−∑(i)

1

1

( 1)3

n

n n

+∞

=

−∑ (j)1

3( 1)

1n

n n

∞

=

−+∑

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(a)1

3 22 1

n

n

nn

∞

=

+ ÷−∑ (b)

1 100

n

n

n∞

=

÷∑

(c)1 5

n

n

n∞

=∑

Use com arison test to determine whetherthe series converges. If the test isinconclusive, then say so.

(a)2

11n n

∞

= −∑ (b)2

1

1n n

∞

= −∑(c)

1

12 3 n

n

∞

= +∑ (d)1

1( 2)n n n

∞

= +∑

(e) 21

12n n n

∞

= −∑

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Absolute and conditional!lassify the series as convergeabsolutely, converge conditionallyor diverges

(a) 2 3 4

1 1 1 1 13 3 3 3 3 n

− − − −+ + " "

(b) 21

cosn

nn

∞

=

∑(c)

1 1 1 11

2 3 4 n− − − −+ + −" "

(d) 1

1

1 1 1 11 ( 1)

2 3 4

n

n n

∞+

=

+ +− − = −

∑"

(e)1

51 2

( 1) n

n n

+∞

=

−∑

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Ratio test for absoluteconvergenceUse ratio test for absolute convergence todetermine whether the series convergesabsolutely. If the test is inconclusive, thensay so.

(a) 2 11

( 10)4 ( 1)

n

n

n n

∞

+

=− +∑ (b)

2

14( 1) 2

n

n

n

n∞

=+−∑

Taylor and Maclaurin series

#ind the $aylor series for 2 xe at x = %u to 5 x .

#ind the &aclaurin series for cos xu to 7 x . Hence, evaluate

(a)1

0cos . x dx∫

(b)1 2

0cos . x dx∫

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Radius and interval ofconvergence#ind the radius and interval of convergence for (ada erubahan)

(a)1

( 1) n n

n

xn

∞

=

−∑(b)

1

1

n

n

xn

+∞

=

∑(c)

1

( 3)( 1) , 2 4, 1

n

n

n

x x R

n

∞

=

−− < ≤ =∑