2.(30 points) These are True-False questions. If the answer is True, you
should explain why (concisely). If the answer is False, you should give a counter-
example.
a) ( 15(15 points )) If the series \sum _ { n = 1 } ^ { \infty } a _ { n }∑n=1∞
a
n
converges absolutely, the sequence \left\{ b _ { n } \right\}{b
n
}
is bounded, then the series \sum _ { n = 1 } ^ { \infty } \sin \left( a _ { n } \right) b _ { n }∑
n=1
∞
sin(a
n
)b
n
also converges absolutely.
b) ( 15(15 points )) If the series \sum _ { n = 1 } ^ { \infty } a _ { n }∑
n=1
∞
a
n
converges conditionally and \sum _ { n = 1 } ^ { \infty } b _ { n }∑
n=1
∞
b
n
converges absolutely, then \sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }∑
n=1
∞
a
n
b
n
converges absolutely.
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