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Anvisha [2.4K]
3 years ago
9

Determine if the following series converges or diverges. If it converges determine its sum.∞∑n=0 (−1)n

Mathematics
1 answer:
Tasya [4]3 years ago
3 0

Answer:  The given series neither converges nor diverges.

Step-by-step explanation: We are given to determine whether the following series converges or diverges :

S=\sum_{n=0}^{\infty}(-1)^n.

If the series converges, we are to find its sum.

The given series can be written as :

1,~-1,~1,~-1,~1,~1,~~.~~.~~.

We note that the given series is a geometric one with first term 1 and common ratio given by

r=\dfrac{-1}{1}=\dfrac{1}{-1}=~~.~~.~~.~~=-1.

We know that a geometric series with common ratio r converges if |r| <1  and  diverges if |r| > 1.

Since |r| = 1 for the given series, so the series will neither converge nor diverge.

Thus, the given series neither converges nor diverges.

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Answer:  In the resulting equation:  " a² - 12a + 32 = 0 " ;
________________________________________________________
  The "coefficient" of the "a" term is:  " - 12" . 
________________________________________________________
  The "constant" is:  " 32 " .
________________________________________________________

Explanation:

________________________________________________________

Let:   "a = x² + 4 " .

Given:  (x² + 4)²  + 32  =  12x² + 48  ; 
______________________________________________________

Factor:  "12x² + 48" into " (x² + 4) " ;

"12x² + 48"  =  12 (x² + 4) " ;
______________________________________________________

Given:    (x² + 4)²  + 32  =  12x² + 48 ; 

rewrite as;  "a² + 32 = 12a " ; 

Subtract "12a" from each side of the equation; 

"a² + 32 - 12a = 12a - 12a ;
 
to get:

" a² - 12a + 32 = 0 " .
___________________________________________________
The coefficient of the "a" term;  that is:

The "coefficient" of " -12a" ;  is:  "- 12" . 

The constant is:  "32<span>" .
</span>___________________________________________________
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