Each of these proofs contains at least one flawed hypothesis (or fallacy) whose

Question

Each of these proofs contains at least one flawed hypothesis (or fallacy) whose introduction into the proof leads to a contradictory conclusion, rendering the proof invalid (or fallacious). For each proof, you must identify at least one fallacy and explain in detail why the fallacy leads to a contradiction. Some of the proofs contain multiple fallacies, but you only need to identify one per proof.

P5 Consider the following infinite series S1 = 1-1+1-1+1-1+… S2-1-2 +3-4 5-6+.. S31+2 +3 + 4 +5+6+.. By a process called Cesàro summation, we can represent the sum of the sequence S1 as when taken to infinity Now let us add the sequence S2 to itself linearly 1-23-4+5-67-. 0+1-2+3-4+5-6+- Then we have just shown that 2(S) = S1 and since S,--as shown above then S2 -4by algebra Now let us subtract S2 from S3 linearly 123 +4+5+6+... - 1- 2+3-4+5- 6+ +4+0 +8+0 12+. Then we will simplify by algebra 0+4+0+8+0+12 + … = 4+8+12 + … = 4(1 2 +3+- 4(S3) So by algebra S3- S24(S3). Then 3(S3)S2, S32, and finally by substituting S2-_ from above, S3-- But S3 is the sum of the natural numbers. Thus, the natural numbers sum to 12 rather than diverging to infinity

Explanation / Answer

1) casaro summation fallacy

S = 1-1+1-1+1-1+1-1+1 .....

= 1 - S = 1 -( 1-1+1-1+1-1+1 ..... )

= 1 - S = (1-1+1-1+1-1+1-1 .... = S)

= S = here fallacy is that we did not consider the sign of infinitieth term, so solution has some incompletion

2) S2 + S2

= 1-2+3-4+5-6+7 ..... infinieth term

+ 0+1-2+3-4+5-6+7.....infiniethh term+ infinieth term+1 term

= 1-1+1-1+1-1+1-1............... infinieth term + 1 term

so, here fallacy is that we didnot considerder infinieth + 1 term, which cause an incomplete solution

3) S3- S2 = 4 S3

S3 = 1/3 S2

but the value of S2 is incomplete according to 2nd condion written above.

so, value of S3 calculated is incomplete.

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