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something funny's going on here, and I can't see what :yuck:

For a sequence [tex] {x_n} [/tex] , where each term is non-negative

the series [tex] x_1 + x_2 + ... +x_n + ... [/tex] converges

proof:

it will suffice to show that the sequence of partial sums [tex] {s_n} [/tex] is bounded

where each [tex]s_i = x_1 + ... + x_i [/tex]

when i=1,

[tex] s_1 = x_1 [/tex]

so the result holds true for i=1

let the result be true for all positive numbers up to some k such that

[tex] s_k <= some b [/tex]

now consider [tex] s_{k+1} [/tex].....

[tex] s_{k+1} = s_k + x_{k+1} <= b + x_{k+1} [/tex]

so the result holds true for all k= 1, 2, 3 ....

For a sequence [tex] {x_n} [/tex] , where each term is non-negative

the series [tex] x_1 + x_2 + ... +x_n + ... [/tex] converges

proof:

it will suffice to show that the sequence of partial sums [tex] {s_n} [/tex] is bounded

where each [tex]s_i = x_1 + ... + x_i [/tex]

when i=1,

[tex] s_1 = x_1 [/tex]

so the result holds true for i=1

let the result be true for all positive numbers up to some k such that

[tex] s_k <= some b [/tex]

now consider [tex] s_{k+1} [/tex].....

[tex] s_{k+1} = s_k + x_{k+1} <= b + x_{k+1} [/tex]

so the result holds true for all k= 1, 2, 3 ....

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