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定理:(Dilichlet)XR $\qquad$若$\,a_n\downarrow 0,\,\{t_n\}\,$有界$\small(\;t_n={\displaystyle\underset{k\le n}{\sum}} b_k,\,\small |t_n|\scriptsize\le M),\,$则$\,\sum a_nb_n\,$收敛.G 证:$\;\displaystyle{\small\sum_{k=m+1}^n}a_kb_k{\small=\sum_{k=m+1}^n}a_k(t_k-t_{k-1}){\small=\sum_{k=m}^n}a_kt_k-{\small\sum_{k=m}^{n-1}}a_{k+1}t_k$f $\qquad=a_nb_n-a_{m+1}b_m+{\small\displaystyle\sum_{k=m+1}^{n-1}}(a_k-a_{k+1})t_k$:1w~ $\therefore\quad\displaystyle\bigg|{\small\sum_{k=m+1}^n}a_kb_k\bigg|\le (a_n{\small+}a_{m+1})\underset{\,}{{\small M+\sum_{k=m+1}^{n-1}}}(a_k{\small-}a_{k+1}){\small M=2}a_{m+1}\small M$W@+3 $\qquad\displaystyle\big\{{\small\sum_{k\le n}}a_kb_k\big\}\,$是Cauchy列.$\small\quad\square$`6
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