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KCC's quizzes AQQ255 about a limit at infinite



		

			

									
					    KCC
					
							

			
				
									
											
			
			
				
  •  Analog Employees  

			
			
			    	
			                    
                
				                on Mar 13, 2024 
				
			
			
		


				
		

		    		    
		    
		    	
			
And here is a new quiz coming!


1. Info: the frequent winners should have received some award and certificate. Please warn me if you think you have been forgotten!


2. Quote of the week: "The elevator to success is out of order. You'll have to use the stairs... one step at a time." - Joe Girard.


3. New quiz of the week: AQQ255 about a limit to infinite


Good luck!

										


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[edited by: emassa at 3:52 PM (GMT -4) on 21 Mar 2024]
				

						
			        
		


				
						
        

    
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                on Mar 13, 2024 2:18 PM
            
            
					
      
	
      

		
		
      
					
      (oops! I send through email rather that here.  I repost it here just for a proper way to do it)

      

      

      

      

      

      

      

      

      

      

      

      

      

      

      

      

      

      

        

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        e
        

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      2. 

         Not really rigorously mathematical, but with a numerical confirmation in Excel, that is "engineeringly" good enough. Here how it goes:

        (1+a)^n      = 1 + na  + n(n-1) a^2  /2!   + n(n-1)(n-2)(n-3)  a^3  ? 3! + ... (binomial expansion)
        (1+1/x)^n   = 1 + n/x  + n(n-1) x^-2  /2! +   ...   (replace a by 1/x )
        (1+1/x)^x    =  1  + 1  +  x(x-1) x^-2  /  2!  +  ...  (replace n  by x )
        

        

        (lim  x -> infinity) ((1+1/x)^x)  = 1 + 1 + x^2 x^-2  /2!  +  x^3 x^-3 / 3! +  ...                  
        

                                                                 (  a finite constant is negligeable in front of infinity)
        

        

        

                                                              = e   (Taylor's serie)
        

        

        3. 

      

      

      

      

      
			
      

		
			
	      

		
		
	
      
	
      
		
      
			
		
      
	
      
	
      
		
						
				
				
						
				
				
		
		
      
			
		
      
	
      

      

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Reply

      


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                on Mar 13, 2024 2:18 PM
            
            
					
        
	
        

		
		
        
					
        (oops! I send through email rather that here.  I repost it here just for a proper way to do it)

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

          

        1. 

          e
          

          2. 

        2. 

           Not really rigorously mathematical, but with a numerical confirmation in Excel, that is "engineeringly" good enough. Here how it goes:

          (1+a)^n      = 1 + na  + n(n-1) a^2  /2!   + n(n-1)(n-2)(n-3)  a^3  ? 3! + ... (binomial expansion)
          (1+1/x)^n   = 1 + n/x  + n(n-1) x^-2  /2! +   ...   (replace a by 1/x )
          (1+1/x)^x    =  1  + 1  +  x(x-1) x^-2  /  2!  +  ...  (replace n  by x )
          

          

          (lim  x -> infinity) ((1+1/x)^x)  = 1 + 1 + x^2 x^-2  /2!  +  x^3 x^-3 / 3! +  ...                  
          

                                                                   (  a finite constant is negligeable in front of infinity)
          

          

          

                                                                = e   (Taylor's serie)
          

          

          3. 

        

        

        

        

        
			
        

		
			
	      

		
		
	
        
	
        
		
        
			
		
        
	
        
	
        
		
						
				
				
						
				
				
		
		
        
			
		
        
	
        

        

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