questions about Rudin's summation by parts

Solution 1:

Define $$ A_n=\sum_{k=1}^na_k\quad\text{and}\quad F(x)=\int_a^xf(t)\,\mathrm{d}t $$

Question 1:

Consider summation by parts $$ \sum_{k=m}^na_kb_k =A_nb_n-A_{m-1}b_m-\sum_{k=m}^{n-1}A_k(b_{k+1}-b_k) $$ vs integration by parts $$ \int_a^bf(x)g(x)\,\mathrm{d}x=F(b)g(b)-F(a)g(a)-\int_a^bF(x)g'(x)\,\mathrm{d}x $$ The parallel seems pretty clear.

Question 2:

I'm not sure what you mean by guidelines to derive summation by parts. Basically, summation by parts is just a change of the index of summation. $$ \begin{align} \sum_{k=m}^na_kb_k &=\sum_{k=m}^n(A_k-A_{k-1})b_k\\ &=\sum_{k=m}^nA_kb_k-\sum_{k=m}^nA_{k-1}b_k\\ &=\sum_{k=m}^nA_kb_k-\sum_{k=m-1}^{n-1}A_kb_{k+1}\\ &=A_nb_n+\sum_{k=m}^{n-1}A_kb_k-A_{m-1}b_m-\sum_{k=m}^{n-1}A_kb_{k+1}\\ &=A_nb_n-A_{m-1}b_m+\sum_{k=m}^{n-1}A_k(b_k-b_{k+1}) \end{align} $$

Question 3:

$$ \sum_{k=5}^na_kb_k =A_nb_n-A_4b_5-\sum_{k=5}^{n-1}A_k(b_{k+1}-b_k) $$ It looks reasonable to me. If you're worried about where the summation starts, that only alters $A_n$ by a constant. Consider how much a constant added to $A_n$ changes the value of the summation by parts formula: $$ \begin{align} &(A_n+C)b_n-(A_{m-1}+C)b_m-\sum_{k=m}^{n-1}(A_k+C)(b_{k+1}-b_k)\\ &=A_nb_n-A_{m-1}b_m-\sum_{k=m}^{n-1}A_k(b_{k+1}-b_k)\\ &+C\left(b_n-b_m-\sum_{k=m}^{n-1}(b_{k+1}-b_k)\right)\\ &=A_nb_n-A_{m-1}b_m-\sum_{k=m}^{n-1}A_k(b_{k+1}-b_k)\\ &+C(b_n-b_m-(b_n-b_m))\\ &=A_nb_n-A_{m-1}b_m-\sum_{k=m}^{n-1}A_k(b_{k+1}-b_k) \end{align} $$ $C$ has no effect on the final value of the formula.