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An equality: from one sum to 2 sums.
The 2019 Stack Overflow Developer Survey Results Are InInfinite Geometric SumDonald Knuth and algebraic operations on sumsSums of infinite seriessum of Problem from Olympiad from bookNice equality involving summation, factorials, and $e$Getting from $sum_n=1^inftyfrac2n3^n+1$ to $frac23sum_n=1^inftysum_m=n^inftyfrac13^m$Transformed Sum questionHow to show the equivalence of these two sums?How to prove equality of sum of Legendre symbolsExpected value sum identity
$begingroup$
I have an equality:
$$
ddota_30 = frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
$$
How from $frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)$ we get $left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right) ?$
Because I do not understand where we lost $frac10,75$ in the first sum.
summation
asked Mar 30 at 18:13
PhilipPhilip
917
$endgroup$
add a comment |
$begingroup$
I have an equality:
$$
ddota_30 = frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
$$
How from $frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)$ we get $left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right) ?$
Because I do not understand where we lost $frac10,75$ in the first sum.
summation
asked Mar 30 at 18:13
PhilipPhilip
917
$endgroup$
$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33
add a comment |
$begingroup$
I have an equality:
$$
ddota_30 = frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
$$
How from $frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)$ we get $left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right) ?$
Because I do not understand where we lost $frac10,75$ in the first sum.
summation
asked Mar 30 at 18:13
PhilipPhilip
917
$endgroup$
I have an equality:
$$
ddota_30 = frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
$$
How from $frac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)$ we get $left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right) ?$
Because I do not understand where we lost $frac10,75$ in the first sum.
summation
summation
asked Mar 30 at 18:13
PhilipPhilip
917
asked Mar 30 at 18:13
PhilipPhilip
917
asked Mar 30 at 18:13
PhilipPhilip
917
asked Mar 30 at 18:13
PhilipPhilip
917
asked Mar 30 at 18:13
PhilipPhilip
917
917
$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33
add a comment |
$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33
$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
beginalignfrac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)&= frac43left( sum_k=0^infty left(frac11.06right)^k left( frac34 - frack120right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k left( 1 - frack90right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
endalign
The $frac43$ and the $frac34$ cancels out.
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
beginalignfrac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)&= frac43left( sum_k=0^infty left(frac11.06right)^k left( frac34 - frack120right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k left( 1 - frack90right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
endalign
The $frac43$ and the $frac34$ cancels out.
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
$endgroup$
add a comment |
$begingroup$
beginalignfrac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)&= frac43left( sum_k=0^infty left(frac11.06right)^k left( frac34 - frack120right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k left( 1 - frack90right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
endalign
The $frac43$ and the $frac34$ cancels out.
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
$endgroup$
add a comment |
$begingroup$
beginalignfrac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)&= frac43left( sum_k=0^infty left(frac11.06right)^k left( frac34 - frack120right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k left( 1 - frack90right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
endalign
The $frac43$ and the $frac34$ cancels out.
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
$endgroup$
beginalignfrac10.75left( sum_k=0^infty left(frac11.06right)^k left( 1 - frac30+k120right) right)&= frac43left( sum_k=0^infty left(frac11.06right)^k left( frac34 - frack120right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k left( 1 - frack90right) right)
\&=left( sum_k=0^infty left(frac11.06right)^k - frac190 sum_k=0^infty kleft(frac11.06right)^k right)
endalign
The $frac43$ and the $frac34$ cancels out.
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
answered Mar 30 at 18:22
Siong Thye GohSiong Thye Goh
104k1468120
104k1468120
add a comment |
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$begingroup$
probably worth noting that $0.75 times 120=90$ and that $120-30=90$
$endgroup$
– Henry
Mar 30 at 18:23
$begingroup$
Hint: The product of $frac10.75$ and $left( 1 - frac30+k120right) $ is $frac10.75-frac10.75cdot frac30120-frac10.75cdot frack120=frac10.75cdot frac120120-frac10.75cdot frac30120-frac10.75cdot frack120$ $=frac900.75cdot 120-frack0.75cdot 120=1-frack0.75cdot 120$
$endgroup$
– callculus
Mar 30 at 18:33