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{Edit: as of 3:55 eastern time, made corrections to tex and itex mistakes}

Is this all kosher in terms of demonstrating accuracy and comprehension of the notation [tex]{a_{1} + a_{2}...} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n}[/tex]

So the lower case represents sequences and upper case represents series.

Sequence: [itex]a_{n} = a_{1}, a_{2}, a_{3}, a_{4}, a_{5}... [/itex]

Series : [itex]A_{n} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5}... [/itex]

Sequence: [itex]b_{n} = a_{1}, (a_{1}+ a_{2}), (a_{1}+ a_{2}+ a_{3}), (a_{1}+ a_{2}+ a_{3}+ a_{4}), (a_{1}+ a_{2}+ a_{3}+ a_{4}+a_{5}), .... [/itex]

Sequence: [itex]b_{n} = b_{1}, b_{2}, b_{3}, b_{4}, b_{5}... [/itex]

Series : [itex]B_{n} = b_{1} + b_{2} + b_{3} + b_{4} + b_{5}... [/itex]

So I can say, [itex]b_{5}=A_{5}[/itex]

For concreteness, suppose a common ratio r = 1/2 is introduced, beginning at [itex]a_{1}=1[/itex],

Sequence: [itex]a_{n} = 1, 1/2, 1/4, 1/8, 1/16...[/itex]

Series: [itex]A_{n} = 1 + (1/2) + (1/4) + (1/8) + (1/16)...[/itex]

Sequence: [itex]b_{n} = 1, (1 + 1/2), (1 + 1/2 + 1/4), (1 + 1/2 + 1/4 + 1/8), (1 + 1/2 + 1/4 + 1/8 + 1/16), .... [/itex]

Sequence: [itex]b_{n} = 1, (3/2), (7/4), (15/8), (31/16)...[/itex]

Series: [itex]B_{n} = 1 + (3/2) + (7/4) + (15/8) + (31/16)...[/itex]

So [itex][b_{5}=A_{5}] = [31/16 = 1 + 1/2 + 1/4 + 1/8 + 1/16] = 1.9375[/itex]

[itex]\lim_{n\rightarrow ∞ } A_{n} = \lim_{n\rightarrow ∞ } b_{n}[/itex] , and to be clear,

[itex]\lim_{n\rightarrow ∞ }[/itex] of Series [itex]A_{n} = \lim_{n\rightarrow ∞ }[/itex] of Sequence [itex]b_{n}[/itex]

And sigma notation [itex]\sum_{n=1}^{n} [/itex] are used to represent series, so

[itex]A_{n}= \sum_{n=1}^{n} a_{n} = b_{n}[/itex]

so for example, using subscript [itex]n=5[/itex]

[itex]A_{5}= \sum_{n=1}^{5} a_{n} = b_{5} = 1.9375[/itex]

And given that we are told the infinite series of [itex]A_{∞} = a_{} + a_{} +...a_{∞} = 2[/itex]

Then we can write [itex]A_{∞}= \lim_{n\rightarrow ∞ } A_{n}=\lim_{n\rightarrow ∞ } b_{n}= \sum_{n=1}^{∞} a_{n} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n}[/itex]

And lastly, the series [itex]B_{n}[/itex] has no role in describing the infinite series [itex] a_{1}, a_{2}, a_{3}, a_{4}, a_{5}... [/itex]

Is this all kosher in terms of demonstrating accuracy and comprehension of the notation [tex]{a_{1} + a_{2}...} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n}[/tex]

So the lower case represents sequences and upper case represents series.

Sequence: [itex]a_{n} = a_{1}, a_{2}, a_{3}, a_{4}, a_{5}... [/itex]

Series : [itex]A_{n} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5}... [/itex]

Sequence: [itex]b_{n} = a_{1}, (a_{1}+ a_{2}), (a_{1}+ a_{2}+ a_{3}), (a_{1}+ a_{2}+ a_{3}+ a_{4}), (a_{1}+ a_{2}+ a_{3}+ a_{4}+a_{5}), .... [/itex]

Sequence: [itex]b_{n} = b_{1}, b_{2}, b_{3}, b_{4}, b_{5}... [/itex]

Series : [itex]B_{n} = b_{1} + b_{2} + b_{3} + b_{4} + b_{5}... [/itex]

So I can say, [itex]b_{5}=A_{5}[/itex]

For concreteness, suppose a common ratio r = 1/2 is introduced, beginning at [itex]a_{1}=1[/itex],

Sequence: [itex]a_{n} = 1, 1/2, 1/4, 1/8, 1/16...[/itex]

Series: [itex]A_{n} = 1 + (1/2) + (1/4) + (1/8) + (1/16)...[/itex]

Sequence: [itex]b_{n} = 1, (1 + 1/2), (1 + 1/2 + 1/4), (1 + 1/2 + 1/4 + 1/8), (1 + 1/2 + 1/4 + 1/8 + 1/16), .... [/itex]

Sequence: [itex]b_{n} = 1, (3/2), (7/4), (15/8), (31/16)...[/itex]

Series: [itex]B_{n} = 1 + (3/2) + (7/4) + (15/8) + (31/16)...[/itex]

So [itex][b_{5}=A_{5}] = [31/16 = 1 + 1/2 + 1/4 + 1/8 + 1/16] = 1.9375[/itex]

[itex]\lim_{n\rightarrow ∞ } A_{n} = \lim_{n\rightarrow ∞ } b_{n}[/itex] , and to be clear,

[itex]\lim_{n\rightarrow ∞ }[/itex] of Series [itex]A_{n} = \lim_{n\rightarrow ∞ }[/itex] of Sequence [itex]b_{n}[/itex]

And sigma notation [itex]\sum_{n=1}^{n} [/itex] are used to represent series, so

[itex]A_{n}= \sum_{n=1}^{n} a_{n} = b_{n}[/itex]

so for example, using subscript [itex]n=5[/itex]

[itex]A_{5}= \sum_{n=1}^{5} a_{n} = b_{5} = 1.9375[/itex]

And given that we are told the infinite series of [itex]A_{∞} = a_{} + a_{} +...a_{∞} = 2[/itex]

Then we can write [itex]A_{∞}= \lim_{n\rightarrow ∞ } A_{n}=\lim_{n\rightarrow ∞ } b_{n}= \sum_{n=1}^{∞} a_{n} = \lim_{n\rightarrow ∞ } \sum_{n=1}^{n} a_{n}[/itex]

And lastly, the series [itex]B_{n}[/itex] has no role in describing the infinite series [itex] a_{1}, a_{2}, a_{3}, a_{4}, a_{5}... [/itex]

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