6. Find lim n? +1 or state that the limit Zn" does not exist:7. Find the sum of the series 8 4 2 that the sum does not exist.or state8. Write 0.63 as a fractio...

Find a formula for the nth term of the sequence of partial sums $\left\{S_{n}\right\} .$ Then evaluate lim $S_{n}$ to obtain the value of the series or state that the series diverges.$^{n \rightarrow \infty}$ $$\sum_{k=1}^{\infty} \frac{1}{(k+6)(k+7)}$$

Here in this problem we are given with telescopic cities, which is Sigma equals to one doing tonight. One divide by K plus six times gate. Blessed seven on. We have to find the form lawful sn, which represents the and IT term of this sequence off Parsons some and then finally of evaluate, limit and approaches in for tonight s and to obtain the value of the given series. So festival Albie using the partial fraction by bosh infection, we can write to see these US Sigma key calls to one toe infinite one divide by K plus six minus one, divided by K plus seven. If you have any doubt that how we are able to use partial fraction, you can see section 8.5 of the book would have some idea. No, next we'll calculate the formula for Essen. A central big words to Sigma K equals to one toe in fight one. Divide by K plus six minus one. Divide back A plus seven. No. Next I'll plug in the value of chaos one, then two, then three and so on. So when I plucked years one, I get the one by seven minus. He'll get one by eight then plus natural blood plugging chaos to swell Get one by eight on Dhere I'll get one by night Similarly, it goes on and when I plug chaos and so I get one place one divide by and place six minus one. Divide by n blessed seven If you observe carefully one by h. R and one by every council wrote Similarly one by nine will be cancelled by the next time on one divide by M Plessix will be controlled by the previous term. Finally, we're left with one divide by seven minus one Divide by and blessed seven Next we have evaluate the limit in approaches in for night essence It is a good stew limit and approaches in for night It s in Is this one So one divide with seven Do I minus one divide in place seven simply We have evaluated. This is so limit and approaches in for night one by seven minus limit and approaches one divide by implicit seven Here it is infinite now very plugged in as in for night Here I get the value of this limit us Zito And finally we get the answer s one by seven so we can compute that The given telescopic cities which waas cake sigma keywords to wondering Tonight one divide way K plus six time scape Lis seven is equals 21 by seven. So this is the final answer for the given problem.

So in the given question we have to show that limit and tends to infinity. Listen will converge to four for the series two plus one plus one by two plus one by four plus so on like so here the first time of the city seven is too and the common issue of the cities ah this to buy even that is one by two. So the ability to solve this question is we have to first find the sum of. In terms of the given cities we have to convert it into a formula and then we have to play the limit that is intended to infinity. Right? So we know that some of in terms of cities is given by ain't one minus our power then divided by one minus all. Yes that is the compensation. And there is the first term of the cities. So the summer of anthems as soon as it goes to the first time is too one minus one by two power than divided by one minus one by two. So when you simplify this, we get as soon as it was to four well deployed by one minus one by two power then right? So we have to find LTD intends to infinite S. And so we can apply limit both sides. So we get four multiplied by one minus one by two power. And so as the annual dance to infinite the value of one by two power and will change to judo. Right? So it's then we'll dance to infinite one by 2000 will tend to zero. So we can substitute the values here we get fall into one minus zero, which is a close to four. Like Yeah, as it is blood that the value of limited interest in finance as an X equals two, it will converge to. For right.

So here we have our series up here at the top. Uh It starts off at any cost. One goes off to infinity, it's a sign of and now note that these values are when your calculator is set to radiance, um I have my calculator set to radiance. You can have as a degree two degrees and you will get the same result of whether or not this divergent or convergent But you want to have the same exact values. Um so just to be clear, I'm using my calculator set to radiance. So to calculate the first term and the sequence of partial sums, we just plug in well value one and it goes on. So we have the sign of one uh which is approximately equal to 0.8415. To calculate the next term. Uh In the sequence of partial sums we will take the previous term. Uh and then now plug in the two for n. So we have the 0.8415 plus the sign of two, which is approximately to 1.7508 Remember we wanted to four decimal places? The third term in the sequence is going to be the previous number plus the sign of now. three, Ah which is approximately equal to 1.8919. Note that this is approximate. Uh And your numbers depending on how you add, it may be slightly off. Uh So you should still get 1.89, but you get you may get 1.89 21 or 1.8917. It's okay. As long as you're approximately close, you should be okay. It just depends on how you add them, whether or not you rounded it before you add it. So the only terms in the sequence is going to be ah 0.8 for 15 is number one. Number two is 1.7508 Number three is 1.8919 Number four is 1.1351 Number five is 0.1762 Number six is negative 0.103 to seven. The seventh term is 0.5538 And the eighth term is 1.5432 Again, your numbers may slightly very, very uh depending on where you rounded, but you should get around that answer. Um Now because we're dealing with a sign, a sign is an oscillating function and right there because it's an oscillating function um and there's it's not one over the sign event or anything like that because it's an oscillating function. Ah We note that ah the series will be divergent and the sequences divergent because of this oscillating feature or nature of the function.

Trying to verify that this serious diversions take a look here. This R is equal to 7/6. She is greater than one, so by the geometric serious Never R is greater than one to serious type purchase.