Evaluating $\lim_{x\to \infty}(\frac{x}{x-1})^x$
Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.
I needed some simple string encryption, so I wrote the following code (with a great deal of "inspiration" from ):      // create and initialize a crypto algorithm     private static SymmetricAlgorithm getAlgorithm(string password) {         SymmetricAlgorithm algorithm = Rijndael.Create();         Rfc2898DeriveBytes rdb = new Rfc2898DeriveBytes(             password, new byte[] {             0x53,0x6f,0x64,0x69,0x75,0x6d,0x20,             // salty goodness             0x43,0x68,0x6c,0x6f,0x72,0x69,0x64,0x65         }         );         algorithm.Padding = PaddingMode.ISO10126;         algorithm.Key = rdb.GetBytes(32);         algorithm.IV = rdb.GetBytes(16);         return algorithm;     }      /*       * encryptString      * provides simple encryption of a string, with a given password      */     public static string encryptString(string clearText, string password) {         SymmetricAlgorithm algorithm = getAlgorithm(password);         byte[] clearBytes = System.Text.Encoding.Unicode.GetBytes(clearText);         MemoryStream ms = new MemoryStream();         CryptoStream cs = new CryptoStream(ms, algorithm.CreateEncryptor(), CryptoStreamMode.Write);         cs.Write(clearBytes, 0, clearBytes.Length);         cs.Close();         return Convert.ToBase64String(ms.ToArray());     }      /*      * decryptString      * provides simple decryption of a string, with a given password      */     public static string decryptString(string cipherText, string password) {         SymmetricAlgorithm algorithm = getAlgorithm(password);         byte[] cipherBytes = Convert.FromBase64String(cipherText);         MemoryStream ms = new MemoryStream();         CryptoStream cs = new CryptoStream(ms, algorithm.CreateDecryptor(), CryptoStreamMode.Write);         cs.Write(cipherBytes, 0, cipherBytes.Length);         cs.Close();                     return System.Text.Encoding.Unicode.GetString(ms.ToArray());     }   The code appears to work fine, except that when decrypting data with an incorrect key, I get a CryptographicException - "Padding is invalid and cannot be removed" - on the cs.Close() line in decryptString.   example code:      string password1 = "password";     string password2 = "letmein";     string startClearText = "The quick brown fox jumps over the lazy dog";     string cipherText = encryptString(startClearText, password1);     string endClearText = decryptString(cipherText, password2);     // exception thrown   My question is, is this to be expected? 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